1 Introduction

Integrable systems of equations that possess sufficiently large number of conservation laws and give rise to multiple soliton solutions play a major role in theoretical physics and in propagation of waves. The work on integrable systems of equations is flourishing because these systems have richer phenomena in scientific applications than the regular systems.

An evolution equation is defined to be integrable in symmetry sense if it admits infinitely many symmetries. Integrable systems are nonlinear differential equations which can be solved analytically. Exactly solvable models and integrable evolution equations in nonlinear science play an essential role in many branches of science and engineering. The useful findings in integrable systems of equations have stimulated much research activity.

The study of constructing integrable systems of equations by using methods, such as recursion operator, symmetries, and bi-Hamiltonian, is an interesting topic of growing interest and has gained large interest recently. Magri [5] studied the connection between conservation laws and symmetries from the geometric point of view, where he proved that some systems admitted two distinct but compatible Hamiltonian structures, now known as bi-Hamiltonian system [2, 3, 15].

In recent years, studies on fifth-order systems of two-component nonlinear evolution equations have received considerable attention [4, 11, 12]. Multi-component generalizations of fifth-order Kaup–Kupershmidt equation

$$\begin{aligned} u_t=u_{5x}+10uu_{3x}+25u_{x}u_{xx}+20u^2u_{x}, \end{aligned}$$
(1)

Sawada–Kotera equation

$$\begin{aligned} u_t=u_{5x}+5uu_{3x}+ 5u_{x}u_{xx}+5u^2u_{x}, \end{aligned}$$
(2)

and Kupershmidt equation

$$\begin{aligned} u_t= & {} u_{5x}+5u_{x}u_{3x}+5u_{xx}^2-5u^2u_{3x}\nonumber \\&-\,20uu_{x}u_{xx}-5u_{x}^3+5u^4u_{x}. \end{aligned}$$
(3)

have been the subject of systematic integrability study. Among these, only five homogeneous systems of two-component cases have been found [7, 9, 12] so far. Here we mention papers pertaining to multi-component generalizations of fifth-order systems only. For the other integrable systems and their properties, we refer the readers to the useful papers [1, 4, 6, 8, 13] and the some of the references therein.

So far the only known integrable systems of fifth-order two-component equations are as follows

$$\begin{aligned} \left( \begin{array}{l} u\\ v\\ \end{array}\right) _{t}= & {} \left( \begin{array}{c} -\frac{5}{3}u_{5x} -10vv_{3x}+10uu_{3x}+ 25u_{x}u_{xx} - 15v_{x}v_{xx} -12u^2u_{x}\\ +6v^2u_{x}+12uvv_{x}-6v^2v_{x}\\ 15v_{5x} -10vu_{3x}-30uv_{3x}-35v_{x}u_{xx} +30v_{x}v_{xx} -45u_{x}v_{xx}\\ +6v^2u_{x}-6v^2v_{x} + 12uvu_{x} + 12u^2v_{x} \end{array} \right) , \end{aligned}$$
(4)
$$\begin{aligned} \left( \begin{array}{l} u\\ v\\ \end{array}\right) _{t}= & {} \left( \begin{array}{c} u_{5x} + 10uu_{3x} + 25u_{x}u_{xx}+20u^2u_{x}+v^2v_{x}\\ u_{3x}v + u_{xx}v_{x}+8uvu_{x}+4u^2v_{x} \end{array} \right) , \end{aligned}$$
(5)
$$\begin{aligned} \left( \begin{array}{l} u\\ v\\ \end{array}\right) _{t}= & {} \left( \begin{array}{c} -\frac{1}{8}u_{5x}-2 uu_{3x}-2u_{x}u_{xx}-\frac{32}{5}u^2u_{x}+v_{x}\\ \frac{9}{8}v_{5x} + 6uv_{3x} + 6u_{x}v_{xx} + 4u_{xx}v_{x}+ \frac{32}{5}u^2v_{x} \end{array} \right) , \end{aligned}$$
(6)
$$\begin{aligned} \left( \begin{array}{l} u\\ v\\ \end{array}\right) _{t}= & {} \left( \begin{array}{c} u_{5}+\frac{5}{2}v_{5}+6u_{3}u+18u_{3}v+12v_{3}u+42v_{3}v+12u_{2}u_{1}+24u_{2}v_{1}+21v_{2}u_{1}\\ +42v_{2}v_{1}+\frac{54}{5}u_{1}u^2+\frac{108}{5}u_{1}uv-18u_{1}v^2+\frac{72}{5}v_{1}u^2-\frac{72}{5}v_{1}uv-144v_{1}v^2\\ \frac{5}{4}u_{5}+\frac{7}{2}v_{5}+3u_{3}u+6v_{3}u-6v_{3}v+\frac{3}{2}u_{2}u_{1}-6u_{2}v_{1}-3v_{2}u_{1}-33v_{2}v_{1}\\ +\frac{36}{5}u_{1}v^2-\frac{18}{5}v_{1}u^2-\frac{36}{5}v_{1}uv+\frac{126}{5}v_{1}v^2 \end{array} \right) , \end{aligned}$$
(7)

and

$$\begin{aligned} \left( \begin{array}{l} u\\ v\\ \end{array}\right) _{t}= \left( \begin{array}{c} u_{5x}-30vv_{4x}+5u_{x}u_{3x}-5u^{2}u_{3x}+15v^{2}u_{3x}-75v_{x}v_{3x}\\ +60uvv_{3x} +90v^{2}v_{3x}+5u_{xx}^{2}-20uu_{x}u_{xx}+60vv_{x}u_{xx}-45v_{xx}^{2}\\ +90vu_{x}v_{xx} +90uv_{x}v_{xx}+540vv_{x}v_{xx}+30u^{2}vv_{xx}-180uv^{2}v_{xx}\\ -90v^{3}v_{xx}-5u_{x}^{3}+45u_{x}v_{x}^{2}+60uvu_{x}v_{x}-180v^{2}u_{x}v_{x} +5u^{4}u_{x} \\ -90u^{2}v^{2}u_{x} +45v^{4}u_{x}+180v_{x}^{3}+30u^{2}v_{x}^{2} -360uvv_{x}^{2} -270v^{2}v_{x}^{2} \\ -60u^{3}vv_{x}+180uv^{3}v_{x} \\ -9v_{5x}+10vu_{4x}+25v_{x}u_{3x}+20uvu_{3x}+30v^{2}u_{3x}+15u_{x}v_{3x}\\ +90v_{x}v_{3x} +15u^{2}v_{3x}+15v^{2}v_{3x}+30u_{xx}v_{xx}+50vu_{x}u_{xx}\\ -10u^{2}vu_{xx}+50uv_{x}u_{xx} +60vv_{x}u_{xx}+60uv^{2}u_{xx}+30v^{3}u_{xx}\\ +90v_{xx}^{2} +60uu_{x}v_{xx}+60vv_{x}v_{xx} +45u_{x}^{2}v_{x}-20uvu_{x}^{2}\\ +60v^{2}u_{x}^{2}-10u^{2}u_{x}v_{x} +90v^{2}u_{x}v_{x} -20u^{3}vu_{x} +120uvu_{x}v_{x}\\ +60uv^{3}u_{x}+15v_{x}^{3} -5u^{4}v_{x} +90u^{2}v^{2}v_{x} -45v^{4}v_{x} \end{array} \right) . \end{aligned}$$
(8)

Bi-Hamiltonian structures and recursion operators for the aforementioned systems are discussed in [5, 7, 9–14] and in some of the references therein. Systems (4) and (5) admit a reduction \(v=0\) to the Kaup–Kupershmidt eqnarray. By setting \(v=0\), system (7) reduces to the Sawada–Kotera equation. By setting \(v=0\) the well-known Kupershmidt equation is an obvious reduction of system (8).

2 New homogeneous fifth-order integrable systems

In the literature, all of classified integrable systems are second- and third-order generalization of the KdV and Burgers equations, or equations related to the KdV and Burgers equations. In the case of fifth-order systems, because of the very big number of arbitrary terms that must be considered, the act of classification of such systems is very complicated. Motivated by some existing examples of bi-Hamiltonian two-component generalization of fifth-order equations, we considered a narrow class of fifth-order two-component systems with specific Jordan matrix for integrability.

From a practical point of view, we observed that in second- and third- order integrable systems, when there is a 2-homogeneous integrable equation in a specific Jordan form, then there is certainly at least one 1, 0-homogeneous system in that Jordan form. Then using the sense of 2-homogeneous fifth-order systems introduced in [5, 7, 9–14], we aim to develop new integrable 1, 0-homogeneous systems in the same Jordan form. Our analysis found four new integrable systems, where some of these systems allow us to write Magri schemes which contain the new systems proving it complete integrability. In what follows, we introduce the new fifth- order two-component systems with the form

$$\begin{aligned} \left( \begin{array}{l} u\\ v\\ \end{array}\right) _{t}= & {} \left( \begin{array}{c} 4u_5+5v_5+20u_4u_1+10u_4v_1+40u_1v_4+20v_4v_1+20u_3u_2-40u_3u_1^2\\ +140u_3u_1v_1+70u_3v_2+80u_3v_1^2+40u_2^2u_1+80u_2^2v_1-80u_2u_1^3\\ +360u_2u_1^2v_1+400u_2u_1v_2+600u_2u_1v_1^2+70u_2v_3+260u_2v_2v_1+200u_2v_1^3\\ +24u_1^5-240u_1^4v_1-160u_1^3v_2+360u_1^3v_1^2+40u_1^2v_3+720u_1^2v_2v_1\\ +1200u_1^2v_1^3+220u_1v_3v_1+370u_1v_2^2+1200u_1v_2v_1^2+600u_1v_1^4+110v_3v_2\\ +100v_3v_1^2+200v_2^2v_1+400v_2v_1^3\\ 10u_5+14v_5-40u_4u_1-20u_4v_1-20u_1v_4+20u_3u_2-40u_3u_1^2\\ -40u_3u_1v_1 +100u_3v_2-100u_3v_1^2-200u_2^2u_1-40u_2^2v_1\\ +160u_2u_1^3-720u_2u_1^2v_1-560u_2u_1v_2-1200u_2u_1v_1^2+100u_2v_3\\ -400u_2v_2v_1-400u_2v_1^3+600u_1^4v_1+80u_1^3v_2+1200u_1^3v_1^2-200u_1^2v_3\\ -360u_1^2v_2v_1+360u_1^2v_1^3-380u_1v_3v_1-320u_1v_2^2-600u_1v_2v_1^2\\ -240u_1v_1^4-10v_4v_1+140v_3v_2-320v_3v_1^2-370v_2^2v_1-200v_2v_1^3+24v_1^5 \end{array}\right) , \end{aligned}$$
(9)
$$\begin{aligned} \left( \begin{array}{l} u\\ v\\ \end{array}\right) _{t}= & {} \left( \begin{array}{c} 4u_{5x}+4v_{4x}v+20u_{3x}u_{x}-20u_{3x}u^2-8u_{3x}v^2+16v_{3x}v_{x}-8v_{3x}uv+20u_{xx}^2\\ -80u_{xx}u_{x}u-8u_{xx}v_{x}v-6u_{xx}uv^2+12v_{xx}^2-12v_{xx}u_{x}v-24v_{xx}v_{x}u\\ -4v_{xx}u^2v-8v_{xx}v^3-20u_{x}^3-12u_{x}^2v^2-12u_{x}v_{x}^2-8u_{x}v_{x}uv+20u_{x}u^4\\ +24u_{x}u^2v^2+u_{x}v^4-4v_{x}^2u^2-12v_{x}^2v^2+8v_{x}u^3v+10v_{x}uv^3\\ -2u_{4x}v+4u_{3x}v_{x}+2u_{3x}uv-2v_{3x}v^2-10u_{xx}u_{x}v-4u_{xx}v_{x}u+8u_{xx}u^2v\\ +4u_{xx}v^3-2v_{xx}v_{x}v+6v_{xx}uv^2+12u_{x}^2v_{x}+16u_{x}^2uv-16u_{x}v_{x}u^2\\ +6u_{x}v_{x}v^2-8u_{x}u^3v+8u_{x}uv^3+4v_{x}^3-6v_{x}^2uv+4v_{x}u^4+5v_{x}v^4\\ \end{array}\right) , \end{aligned}$$
(10)
$$\begin{aligned} \left( \begin{array}{l} u\\ v\\ \end{array}\right) _{t}= & {} \left( \begin{array}{c} {u}_{5x}+{v}_{5x}+2{u}_{x}{u}_{4x}-2{v}_{x}{u}_{4x}+6{u}_{x}{v}_{4x}-6{v}_{x}{v}_{4x} -16{u}_{xx}{u}_{3x}\\ -4{v}_{xx}{u}_{3x} -54{u}_{x}^{2}{u}_{3x} -20{u}_{x}{v}_{x}{u}_{3x}-6{v}_{x}^{2}{u}_{3x}-4{u}_{xx}{v}_{3x} \\ -16{v}_{xx}{v}_{3x}-22{u}_{x}^{2}{v}_{3x} -52{u}_{x}{v}_{x}{v}_{3x} -6{v}_{x}^{2}{v}_{3x} -52{u}_{x}{u}_{xx}^{2} -4{v}_{x}{u}_{xx}^{2}\\ -32{u}_{x}{u}_{xx}{v}_{xx}-16{v}_{x}{u}_{xx}{v}_{xx} -12{u}_{x}^{3}{u}_{xx} +4{u}_{x}^{2}{v}_{x}{u}_{xx} -4{u}_{x}{v}_{x}^{2}{u}_{xx}\\ +12{v}_{x}^{3}{u}_{xx}-44{u}_{x}{v}_{xx}^{2}-12{v}_{x}{v}_{xx}^{2} -36{u}_{x}^{3}{v}_{xx} +12{u}_{x}^{2}{v}_{x}{v}_{xx}\\ -12{u}_{x}{v}_{x}^{2}{v}_{xx} +36{v}_{x}^{3}{v}_{xx}+72{u}_{x}^{5}+96{u}_{x}^{4}{v}_{x} +176{u}_{x}^{3}{v}_{x}^{2} +96{u}_{x}^{2}{v}_{x}^{3}\\ +72{u}_{x}{v}_{x}^{4}\\ {u}_{5x}+{v}_{5x}-6{u}_{x}{u}_{4x}+6{v}_{x}{u}_{4x}-2{u}_{x}{v}_{4x}+2{v}_{x}{v}_{4x}-16{u}_{xx}{u}_{3x} \\ -4{v}_{xx}{u}_{3x} -6{u}_{x}^{2}{u}_{3x} -52{u}_{x}{v}_{x}{u}_{3x} -22{v}_{x}^{2}{u}_{3x}-4{u}_{xx}{v}_{3x}\\ -16{v}_{xx}{v}_{3x} -6{u}_{x}^{2}{v}_{3x} -20{u}_{x}{v}_{x}{v}_{3x}-54{v}_{x}^{2}{v}_{3x} -12{u}_{x}{u}_{xx}^{2}\\ -44{v}_{x}{u}_{xx}^{2} -16{u}_{x}{u}_{xx}{v}_{xx} -32{v}_{x}{u}_{xx}{v}_{xx} +36{u}_{x}^{3}{u}_{xx}\\ -12{u}_{x}^{2}{v}_{x}{u}_{xx} +12{u}_{x}{v}_{x}^{2}{u}_{xx} -36{v}_{x}^{3}{u}_{xx} -4{u}_{x}{v}_{xx}^{2}-52{v}_{x}{v}_{xx}^{2} \\ +12{u}_{x}^{3}{v}_{xx} -4{u}_{x}^{2}{v}_{x}{v}_{xx} +4{u}_{x}{v}_{x}^{2}{v}_{xx} -12{v}_{x}^{3}{v}_{xx} +72{u}_{x}^{4}{v}_{x}\\ +96{u}_{x}^{3}{v}_{x}^{2} +176{u}_{x}^{2}{v}_{x}^{3} +96{u}_{x}{v}_{x}^{4}+72{v}_{x}^{5} \end{array}\right) , \end{aligned}$$
(11)

and

$$\begin{aligned} \left( \begin{array}{l} u\\ v\\ \end{array}\right) _{t}= \left( \begin{array}{c} {u}_{5x}+{v}_{5x}+{u}_{x}{u}_{4x}-{v}_{x}{u}_{4x}+3{u}_{x}{v}_{4x}-3{v}_{x}{v}_{4x}+7{u}_{xx}{u}_{3x}\\ +13{v}_{xx}{u}_{3x} -36{u}_{x}^{2}{u}_{3x} -20{u}_{x}{v}_{x}{u}_{3x} -24{v}_{x}^{2}{u}_{3x}+13{u}_{xx}{v}_{3x}\\ +7{v}_{xx}{v}_{3x} -28{u}_{x}^{2}{v}_{3x}-28{u}_{x}{v}_{x}{v}_{3x}-24{v}_{x}^{2}{v}_{3x} -28{u}_{x}{u}_{xx}^{2}\\ -16{v}_{x}{u}_{xx}^{2}-8{u}_{x}{u}_{xx}{v}_{xx} -64{v}_{x}{u}_{xx}{v}_{xx}-24{u}_{x}^{3}{u}_{xx}+8{u}_{x}^{2}{v}_{x}{u}_{xx}\\ -8{u}_{x}{v}_{x}^{2}{u}_{xx} +24{v}_{x}^{3}{u}_{xx}+4{u}_{x}{v}_{xx}^{2}-48{v}_{x}{v}_{xx}^{2}-72{u}_{x}^{3}{v}_{xx}\\ +24{u}_{x}^{2}{v}_{x}{v}_{xx} -24{u}_{x}{v}_{x}^{2}{v}_{xx} +72{v}_{x}^{3}{v}_{xx} +72{u}_{x}^{5}+96{u}_{x}^{4}{v}_{x}\\ +176{u}_{x}^{3}{v}_{x}^{2} +96{u}_{x}^{2}{v}_{x}^{3} +72{u}_{x}{v}_{x}^{4}\\ {u}_{5x}+{v}_{5x}-3{u}_{x}{u}_{4x}+3{v}_{x}{u}_{4x}-{u}_{x}{v}_{4x}+{v}_{x}{v}_{4x}+7{u}_{xx}{u}_{3x}\\ +13{v}_{xx}{u}_{3x}-24{u}_{x}^{2}{u}_{3x} -28{u}_{x}{v}_{x}{u}_{3x}-28{v}_{x}^{2}{u}_{3x}+13{u}_{xx}{v}_{3x}\\ +7{v}_{xx}{v}_{3x} -24{u}_{x}^{2}{v}_{3x}-20{u}_{x}{v}_{x}{v}_{3x} -36{v}_{x}^{2}{v}_{3x} -48{u}_{x}{u}_{xx}^{2}\\ +4{v}_{x}{u}_{xx}^{2} -64{u}_{x}{u}_{xx}{v}_{xx}-8{v}_{x}{u}_{xx}{v}_{xx}+72{u}_{x}^{3}{u}_{xx} -24{u}_{x}^{2}{v}_{x}{u}_{xx}\\ +24{u}_{x}{v}_{x}^{2}{u}_{xx} -72{v}_{x}^{3}{u}_{xx}-16{u}_{x}{v}_{xx}^{2}-28{v}_{x}{v}_{xx}^{2}+24{u}_{x}^{3}{v}_{xx} \\ -8{u}_{x}^{2}{v}_{x}{v}_{xx} +8{u}_{x}{v}_{x}^{2}{v}_{xx}-24{v}_{x}^{3}{v}_{xx} +72{u}_{x}^{4}{v}_{x}+96{u}_{x}^{3}{v}_{x}^{2} \\ +176{u}_{x}^{2}{v}_{x}^{3} +96{u}_{x}{v}_{x}^{4}+72{v}_{x}^{5} \end{array}\right) . \end{aligned}$$
(12)

To find the second set of systems, we use a classification that we restricted to the case \(\lambda =0\) homogeneous symmetrically coupled systems. We determine all equations of the form as

$$\begin{aligned} \left( \begin{array}{l} u\\ v\\ \end{array}\right) _{t}= \left( \begin{array}{c} A[u_x,v_x] \\ A[v_x,u_x] \end{array}\right) . \end{aligned}$$
(13)

with the class of two-component 0-homogeneous symmetrically coupled systems with undetermined constant coefficients \(\gamma _{i}\) have the form

$$\begin{aligned} A= & {} {\gamma }_{1} u_{5x}+{\gamma }_{2} v_{5x}+{\alpha }_{1}u_{x}u_{4x}+{\alpha }_{2}v_{x}u_{4x}\nonumber \\&+\,{\alpha }_{3}u_{x}v_{4x}+{\alpha }_{4}v_{x}v_{4x}+{\alpha }_{5}u_{xx}u_{3x}\nonumber \\&+\,{\alpha }_{6}v_{xx}u_{3x}+{\alpha }_{7}u_{x}^2u_{3x}\nonumber \\&+\,{\alpha }_{8}u_{x}v_{x}u_{3x}+{\alpha }_{9}v_{x}^2u_{3x}\nonumber \\&+\,{\alpha }_{10}u_{xx}v_{3x}+{\alpha }_{11}v_{xx}v_{3x}\nonumber \\&+\,{\alpha }_{12}u_{x}^2v_{3x}+{\alpha }_{13}u_{x}v_{x}v_{3x}+{\alpha }_{14}v_{x}^2v_{3x}\nonumber \\&+\,{\alpha }_{15}u_{x}u_{xx}^2+{\alpha }_{16}v_{x}u_{xx}^2\nonumber \\&+\,{\alpha }_{17}u_{x}v_{xx}u_{xx}+{\alpha }_{18}v_{x}v_{xx}u_{xx}\nonumber \\&+\,{\alpha }_{19}u_{x}^3u_{xx}+{\alpha }_{20}u_{x}^2v_{x}u_{xx}\nonumber \\&+\,{\alpha }_{21}u_{x}v_{x}^2u_{xx}\nonumber \\&+\,{\alpha }_{22}v_{x}^3u_{xx}+{\alpha }_{23}u_{x}v_{xx}^2\nonumber \\&+\,{\alpha }_{24}v_{x}v_{xx}^2+{\alpha }_{25}u_{x}^3v_{xx}\nonumber \\&+\,{\alpha }_{26}u_{x}^2v_{x}v_{xx}\nonumber \\&+\,{\alpha }_{27}u_{x}v_{x}^2v_{xx}\nonumber \\&+\,{\alpha }_{28}v_{x}^3v_{xx}\nonumber \\&+\,{\alpha }_{29}u_{x}^5+{\alpha }_{30}u_{x}^4v_{x}\nonumber \\&+\,{\alpha }_{31}u_{x}^3v_{x}^2+{\alpha }_{32}u_{x}^2v_{x}^3\nonumber \\&+\,{\alpha }_{33}u_{x}v_{x}^4+{\alpha }_{34}v_{x}^5 \end{aligned}$$
(14)

possessing an admissible generator of form (13) with the main matrix of these systems is \( \left( \begin{array}{cc} {\gamma }_{1} &{}\quad {\gamma }_{2} \\ {\gamma }_{2} &{}\quad {\gamma }_{1} \end{array}\right) \). By a linear change of variables, the matrix (13) can be reduced to following canonical Jordan form \(\left( \begin{array}{cc} {\gamma }_{1} + {\gamma }_{2}&{} \quad 0 \\ 0 &{}\quad {\gamma }_{1} - {\gamma }_{2} \end{array}\right) . \) Because of properties of symmetric systems, we will restrict our attention to \( \gamma _2\le \gamma _1,~\gamma _{1,2}=0,1\). Similarly, we will deal with two canonical Jordan form \(\left( \begin{array}{cc} 1 &{}\quad 0 \\ 0 &{}\quad 1 \end{array}\right) \) and \( \left( \begin{array}{cc} 1 &{}\quad 0 \\ 0 &{}\quad 0 \end{array}\right) \). Imposing compatibility condition among the classes of systems and an arbitrary seventh-order 0-homogeneous, we obtain a system of equations among the undetermined constants. If we separate out the coefficients of powers of u and v in this equation, then in some condition the coefficients of \(u_{nx}^m*v_{n^{\prime }x}^{m\prime }\) all vanish identically. Solutions of the compatibility condition are given in the following theorems.

Theorem 2.1

A coupled fifth-order system of two-component evolution equations of the forms (13) and (14) that possesses a seventh-order generalized symmetry of form (13) with \(\gamma _1=\gamma _2=1\) has a lower order symmetry or can be transformed by a linear change of variables to one of the following two systems (11) and (12).

Theorem 2.2

Every coupled fifth-order system of two-component evolution equations of form (13) and (14) that possesses a seventh-order generalized symmetry of form (13) with \(\gamma _1=1, \gamma _2=0\) has a lower order symmetry.

2.1 Integrability of the system (9)

System (9) possesses a symplectic operator as

$$\begin{aligned} S = \left( \begin{array}{cc} 2D_x&{}\quad D_x \\ D_x&{} \quad 2D_x \end{array}\right) . \end{aligned}$$
(15)

Second Hamiltonian or symplectic operator for this system is an open question for us.

2.2 Integrability of the system (10)

Our main concern now is to show the integrability of the system (2.2). To achieve this goal, we set

$$\begin{aligned} R = \left( \begin{array}{cc} R _{1}&{}\quad R _{2}\\ R_{3}&{}\quad R_4 \end{array}\right) \end{aligned}$$
(16)

where

$$\begin{aligned} R_1= & {} \alpha _{1} D_x^6 + \alpha _{2} D_x^4 + \alpha _{3} D_x^3 + \alpha _{4} D_x^2 \\&+\, \alpha _{5} D_x + \alpha _{6}+ \alpha _{01} D_x^{-1} \alpha _{07} + \alpha _{02} D_x^{-1} \alpha _{05} \\ R_2= & {} \alpha _{7} D_x^5 + \alpha _{8} D_x^4 + \alpha _{9} D_x^3 + \alpha _{10} D_x^2 \\&+\, \alpha _{11} D_x + \alpha _{12} + \alpha _{01} D_x^{-1} \alpha _{08} + \alpha _{02} D_x^{-1} \alpha _{06} \\ R_3= & {} \alpha _{13} D_x^5 + \alpha _{14} D_x^4 + \alpha _{15} D_x^3 + \alpha _{16} D_x^2 \\&+\, \alpha _{17} D_x + \alpha _{18}+ \alpha _{03} D_x^{-1} \alpha _{07} + \alpha _{04} D_x^{-1} \alpha _{05}\\ R_4= & {} \alpha _{19} D_x^4 + \alpha _{20} D_x^3 + \alpha _{21} D_x^2 + \alpha _{22} D_x + \alpha _{23} \\&+\, \alpha _{03} D_x^{-1} \alpha _{08} + \alpha _{04} D_x^{-1} \alpha _{06} \\ \alpha _{1}= & {} 12\\ \alpha _{2}= & {} 72{u}_{1}-72u^2-30v^2 \\ \alpha _{3}= & {} 180{u}_{2}-360{u}_{1}u-18uv^2-60{v}_{1}v \\ \alpha _{4}= & {} 168{u}_{3}-480{u}_{2}u-372{u}_{1}^2-72{u}_{1}u^2\\&-\,126{u}_{1}v^2+108u^4+114u^2v^2\\&-\,48u{v}_{1}v-72{v}_{2}v-72{v}_{1}^2+12v^4 \\ \alpha _{5}= & {} 72{u}_{4}-360{u}_{3}u-756{u}_{2}{u}_{1}-108{u}_{2}u^2\\&-\,126{u}_{2}v^2-216{u}_{1}^2u\\&+\,648{u}_{1}u^3+342{u}_{1}uv^2-180{u}_{1}{v}_{1}v\\&+\,18u^3v^2+180u^2{v}_{1}v-72u{v}_{2}v-72u{v}_{1}^2\\&+\,9uv^4-48{v}_{3}v-144{v}_{2}{v}_{1}+54{v}_{1}v^3 \\ \alpha _{6}= & {} 12{u}_{5}-144{u}_{4}u-276{u}_{3}{u}_{1}-36{u}_{3}u^2\\&-\,36{u}_{3}v^2-180{u}_{2}^2-456{u}_{2}{u}_{1}u\\&+\,456{u}_{2}u^3+210{u}_{2}uv^2-96{u}_{2}{v}_{1}v-72{u}_{1}^3\\&+\,888{u}_{1}^2u^2+132{u}_{1}^2v^2+108{u}_{1}u^2v^2\\&+\,288{u}_{1}u{v}_{1}v-84{u}_{1}{v}_{2}v-84{u}_{1}{v}_{1}^2\\&+\,18{u}_{1}v^4-48u^6-84u^4v^2+48u^3{v}_{1}v\\&+\,156u^2{v}_{2}v+156u^2{v}_{1}^2-30u^2v^4-72u{v}_{3}v\\&-\,216u{v}_{2}{v}_{1}+78u{v}_{1}v^3-12{v}_{4}v-48{v}_{3}{v}_{1}\\&-\,36{v}_{2}^2+24{v}_{2}v^3+36{v}_{1}^2v^2 \\ \alpha _{7}= & {} 12v \\ \end{aligned}$$
$$\begin{aligned} \alpha _{8}= & {} -24uv+60{v}_{1} \\ \alpha _{9}= & {} -48{u}_{1}v-24u^2v-96u{v}_{1}+120{v}_{2}-30v^3 \end{aligned}$$
$$\begin{aligned} \alpha _{10}= & {} -60{u}_{2}v-72{u}_{1}uv-144{u}_{1}{v}_{1}+48u^3v-72u^2{v}_{1}\\&-\,144u{v}_{2}+42uv^3+120{v}_{3}-150{v}_{1}v^2 \\ \alpha _{11}= & {} -72{u}_{3}v-72{u}_{2}uv-120{u}_{2}{v}_{1}-84{u}_{1}^2v+216{u}_{1}u^2v\\&-\,144{u}_{1}u{v}_{1}-144{u}_{1}{v}_{2}+54{u}_{1}v^3\\&+\,12u^4v+96u^3{v}_{1}-72u^2{v}_{2}\\&+\,30u^2v^3-96u{v}_{3}+156u{v}_{1}v^2+60{v}_{4}\\&-\,162{v}_{2}v^2-192{v}_{1}^2v+12v^5 \\ \alpha _{12}= & {} -48{u}_{4}v-24{u}_{3}uv-72{u}_{3}{v}_{1}-228{u}_{2}{u}_{1}v\\&+\,180{u}_{2}u^2v-72{u}_{2}u{v}_{1}-60{u}_{2}{v}_{2}\\&+\,54{u}_{2}v^3+240{u}_{1}^2uv-84{u}_{1}^2{v}_{1}+24{u}_{1}u^3v\\&+\,216{u}_{1}u^2{v}_{1}-\,72{u}_{1}u{v}_{2}+66{u}_{1}uv^3-48{u}_{1}{v}_{3}\\&+\,114{u}_{1}{v}_{1}v^2-24u^5v+12u^4{v}_{1}\\&+\,48u^3{v}_{2}-42u^3v^3-24u^2{v}_{3}+\,66u^2{v}_{1}v^2\\&-24u{v}_{4}+114u{v}_{2}v^2+144u{v}_{1}^2v-15uv^5+\,12{v}_{5}\\&-78{v}_{3}v^2-276{v}_{2}{v}_{1}v-72{v}_{1}^3+66{v}_{1}v^4 \\ \alpha _{13}= & {} -6v \end{aligned}$$
$$\begin{aligned} \alpha _{14}= & {} 6uv+12{v}_{1} \\ \alpha _{15}= & {} -36{u}_{1}v+30u^2v-12u{v}_{1}+15v^3 \\ \alpha _{16}= & {} -54{u}_{2}v+150{u}_{1}uv\\&+\,84{u}_{1}{v}_{1}-30u^3v-60u^2{v}_{1}+21uv^3+24{v}_{1}v^2 \\ \alpha _{17}= & {} -30{u}_{3}v+144{u}_{2}uv+24{u}_{2}{v}_{1}+78{u}_{1}^2v\\&-\,54{u}_{1}u^2v-180{u}_{1}u{v}_{1}+45{u}_{1}v^3-24u^4v\\&+\,60u^3{v}_{1}-24u^2v^3+30u{v}_{1}v^2\\&+\,18{v}_{2}v^2-30{v}_{1}^2v-6v^5 \\ \alpha _{18}= & {} -6{u}_{4}v+66{u}_{3}uv+36{u}_{3}{v}_{1}\\&+\,90{u}_{2}{u}_{1}v-36{u}_{2}u^2v\\&-\,108{u}_{2}u{v}_{1}+18{u}_{2}v^3+48{u}_{1}^2uv\\&+\,24{u}_{1}^2{v}_{1}-144{u}_{1}u^3v-72{u}_{1}u^2{v}_{1}-72{u}_{1}uv^3\\&+\,18{u}_{1}{v}_{1}v^2+24u^5v+48u^4{v}_{1}-12u^3v^3\\&-\,48u^2{v}_{1}v^2+18u{v}_{2}v^2-42u{v}_{1}^2v-12uv^5\\&+\,6{v}_{3}v^2+30{v}_{2}{v}_{1}v+12{v}_{1}^3-18{v}_{1}v^4 \\ \alpha _{19}= & {} -6v^2 \\ \alpha _{20}= & {} 18uv^2-12{v}_{1}v \\ \alpha _{21}= & {} 12{u}_{1}v^2-6u^2v^2+18u{v}_{1}v-36{v}_{2}v+36{v}_{1}^2+15v^4 \\ \alpha _{22}= & {} 18{u}_{2}v^2+18{u}_{1}uv^2+36{u}_{1}{v}_{1}v-18u^3v^2\\&+\,54u{v}_{2}v-72u{v}_{1}^2-9uv^4-24{v}_{3}v\\&+\,36{v}_{2}{v}_{1}+54{v}_{1}v^3 \\ \alpha _{23}= & {} 18{u}_{3}v^2-30{u}_{2}{v}_{1}v+30{u}_{1}^2v^2-66{u}_{1}u^2v^2-42{u}_{1}u{v}_{1}v\\&+\,12{u}_{1}{v}_{2}v+12{u}_{1}{v}_{1}^2-15{u}_{1}v^4+12u^4v^2+18u^3{v}_{1}v\\&-\,6u^2{v}_{2}v+12u^2{v}_{1}^2-6u^2v^4+18u{v}_{3}v\\&-\,36u{v}_{2}{v}_{1}-27u{v}_{1}v^3\\&-\,6{v}_{4}v+12{v}_{3}{v}_{1}+33{v}_{2}v^3-6{v}_{1}^2v^2-6v^6 \\ \end{aligned}$$
$$\begin{aligned} \alpha _{01}= & {} -24{u}_{5}-120{u}_{3}{u}_{1}+120{u}_{3}u^2\\&+\,48{u}_{3}v^2-120{u}_{2}^2+480{u}_{2}{u}_{1}u+36{u}_{2}uv^2\\&+\,48{u}_{2}{v}_{1}v+120{u}_{1}^3+72{u}_{1}^2v^2-120{u}_{1}u^4-144{u}_{1}u^2v^2\\&+\,48{u}_{1}u{v}_{1}v\\&+\,72{u}_{1}{v}_{2}v+72{u}_{1}{v}_{1}^2-6{u}_{1}v^4-48u^3{v}_{1}v+24u^2{v}_{2}v\\&+\,24u^2{v}_{1}^2+48u{v}_{3}v+144u{v}_{2}{v}_{1}\\&-\,60u{v}_{1}v^3-24{v}_{4}v-96{v}_{3}{v}_{1}\\&-\,72{v}_{2}^2+48{v}_{2}v^3+72{v}_{1}^2v^2 \\ \alpha _{02}= & {} -3{u}_{1} \\ \alpha _{03}= & {} 12{u}_{4}v-12{u}_{3}uv-24{u}_{3}{v}_{1}+60{u}_{2}{u}_{1}v\\&-\,48{u}_{2}u^2v+24{u}_{2}u{v}_{1}-24{u}_{2}v^3-96{u}_{1}^2uv\\&-\,72{u}_{1}^2{v}_{1}+48{u}_{1}u^3v+96{u}_{1}u^2{v}_{1}-48{u}_{1}uv^3\\&-\,36{u}_{1}{v}_{1}v^2-24u^4{v}_{1}-36u{v}_{2}v^2\\&+\,36u{v}_{1}^2v+12{v}_{3}v^2\\&+\,12{v}_{2}{v}_{1}v-24{v}_{1}^3-30{v}_{1}v^4 \\ \alpha _{04}= & {} -3{v}_{1} \\ \alpha _{05}= & {} 8u_4+8v_3v+40u_2u_1-40u_2u^2-4u_2v^2+24v_2v_1\\&-\,16v_2uv-40u_1^2u-8u_1v_1v\\&-\,16v_1^2u-8v_1u^2v-4v_1v^3+8u^5+8u^3v^2+2uv^4 \\ \alpha _{06}= & {} -8u_3v-16u_2uv-8v_2v^2-12u_1^2v\\&+\,8u_1u^2v+4u_1v^3-8v_1^2v+4u^4v+4u^2v^3+v^5 \\ \alpha _{07}= & {} u \\ \alpha _{08}= & {} \frac{v}{2} \\ \end{aligned}$$

This shows that the system (10) passes the integrability test.

2.3 Integrability of system (11)

We proceed as before to show the integrability of the system (11). By change of dependent variables

$$\begin{aligned} u\rightarrow & {} \frac{1}{2}\int (w-z){\mathrm {d}}x \end{aligned}$$
(17)
$$\begin{aligned} v\rightarrow & {} \frac{1}{2}\int (w+z){\mathrm {d}}x \end{aligned}$$
(18)

system (11) can be written in its canonical form as

$$\begin{aligned} \left( \begin{array}{l} u\\ v\\ \end{array}\right) _{t}= \left( \begin{array}{c} w_{5x}-2zz_{4x}-10w_{x}w_{3x}-20w^2w_{3x}-2z^2w_{3x}-8z_{x}z_{3x}\\ -8wzz_{3x} -10w_{xx}^2 -80ww_{x}w_{xx}-8zz_{x}w_{xx} -6z_{xx}^2\\ -12zw_{x}z_{xx} -24wz_{x}z_{xx} +8w^2zz_{xx} +4z^3z_{xx}-20w_{x}^3-12w_{x}z_{x}^2\\ +16wzw_{x}z_{x}+80w^4w_{x} +48w^2z^2w_{x} +4z^4w_{x}+8w^2z_{x}^2\\ +12z^2z_{x}^2 +32w^3zz_{x}+16wz^3z_{x} \\ 4zw_{4x}+4z_{x}w_{3x}-16wzw_{3x}-8z^2z_{3x}-40zw_{x}w_{xx}\\ -16wz_{x}w_{xx} -16w^2zw_{xx} -8z^3w_{xx} -32zz_{x}z_{xx} -12w_{x}^2z_{x} \\ -32wzw_{x}^2 -16w^2w_{x}z_{x} -24z^2w_{x}z_{x} +64w^3zw_{x}+32wz^3w_{x}\\ -8z_{x}^3+16w^4z_{x} +48w^2z^2z_{x} +20z^4z_{x} \end{array}\right) \end{aligned}$$
(19)

Proposition

The infinite hierarchy of the system (19) can be written in two different ways

$$\begin{aligned} \left( \begin{array}{c} w_{t}\\ z_{t} \end{array} \right) = {J} \left( \begin{array}{c} \delta _{w}\\ \delta _{z} \end{array} \right) \int \rho _{1}~ \mathrm {d}x =K \left( \begin{array}{c} \delta _{w}\\ \delta _{z} \end{array} \right) \int \rho _{0} ~\mathrm {d}x \end{aligned}$$
(20)

with the compatible pair of Hamiltonian operators

$$\begin{aligned} {J}= \left( \begin{array}{cc} D_x &{}\quad 0 \\ 0&{} \quad 2D_x \end{array}\right) ,\quad {K} = \left( \begin{array}{cc} {K} _{1}^1&{}\quad {K} _{2}^1\\ {K}_{3}^1&{}\quad {K} _{4}^1 \end{array}\right) \end{aligned}$$
(21)

where

$$\begin{aligned} K_1^1= & {} D_x^7+\omega _1D_x^5+D_x^5\omega _1+\omega _2D_x^3+D_x^3\omega _2\nonumber \\&+\,\omega _3D_x+D_x\omega _3+8w_{x}D_x^{-1}w_t\nonumber \\&+\,8w_tD_x^{-1}w_{x}\nonumber \\ K_2^1= & {} D_x^6\omega _4+D_x^5\omega _5+D_x^4\omega _6+D_x^3\omega _7\nonumber \\&+\,D_x^2\omega _8+D_x\omega _9+\omega _{10}\nonumber \\&+\,8w_{x}D_x^{-1}z_t\nonumber \\&+\,8w_tD_x^{-1}z_{x} \nonumber \\ K_3^1= & {} -\omega _4D_x^6+ \omega _5D_x^5- \omega _6D_x^4\nonumber \\&+\, \omega _7D_x^3- \omega _8D_x^2+ \omega _9D_x-\omega _{10} +8z_{x}D_x^{-1}w_t\nonumber \\&+\,8z_tD_x^{-1}w_{x}\nonumber \\ k_4^1= & {} \omega _{11}D_x^5+D_x^5\omega _{11}+\omega _{12}D_x^3\nonumber \\&+\,D_x^3\omega _{12}+\omega _{13}D_x+D_x\omega _{13} +8z_{x}D_x^{-1}z_t\nonumber \\&+\,8z_tD_x^{-1}z_{x} \end{aligned}$$
(22)

and the coefficients satisfy

$$\begin{aligned} \omega _1= & {} -6w_{x}-12w^2-2z^2 \nonumber \\ \omega _2= & {} 16w_{3x}+40ww_{xx}+8zz_{xx}+58w_{x}^2\nonumber \\&+\,24w^2w_{x}+12z^2w_{x}+\,8z_{x}^2+16wzz_{x}\nonumber \\&+\,72w^4 +40w^2z^2 +18z^4 \nonumber \\ \omega _3= & {} -10w_{5x}-24ww_{4x}-4zz_{4x}-100w_{x}w_{3x}\nonumber \\&-\,24w^2w_{3x}-12z^2w_{3x}\nonumber \end{aligned}$$
$$\begin{aligned}&-\,16z_{x}z_{3x}\nonumber \\&-\,84w_{xx}^2-64ww_{x}w_{xx} -64zz_{x}w_{xx}\nonumber \\&-\,128w^3w_{xx}-64wz^2w_{xx}-12z_{xx}^2\nonumber \\&-\,56zw_{x}z_{xx}-16w^2zz_{xx}\nonumber \\&-152z^3z_{xx} -48w_{x}^3-\,704w^2w_{x}^2-80z^2w_{x}^2\nonumber \\&-\,56w_{x}z_{x}^2-288wzw_{x}z_{x} -96z^4w_{x}\nonumber \\&-\,16w^2z_{x}^2-216z^2z_{x}^2\nonumber \\&-\,64w^3zz_{x} +96wz^3z_{x}\\&-\,128w^6-128w^4z^2-\,96w^2z^4-16z^6\nonumber \\ \omega _4= & {} -4z \nonumber \\ \omega _5= & {} 4z_{x}-16wz\nonumber \\ \omega _6= & {} +48zw_{x}+16wz_{x}+32w^2z \nonumber \\ \omega _7= & {} -72zw_{xx}-32w_{x}z_{x}-160wzw_{x}\nonumber \\&-\,32w^2z_{x}+96z^2z_{x}+128w^3z \nonumber \\ \omega _8= & {} +40zw_{3x}+40z_{x}w_{xx}+192wzw_{xx}\nonumber \\&+208zw_{x}^2+\,96ww_{x}z_{x}-576w^2zw_{x}\nonumber \\&-\,320zz_{x}^2 -128w^3z_{x} -64w^4z \nonumber \\ \omega _9= & {} -8zw_{4x}-128wzw_{3x}-104z^2z_{3x}\nonumber \\&-\,240zw_{x}w_{xx}-96wz_{x}w_{xx}+480w^2zw_{xx}\nonumber \\&-\,96z^3w_{xx} -152zz_{x}z_{xx}+576wz^2z_{xx}-112w_{x}^2z_{x}\nonumber \\&+\,640wzw_{x}^2+192w^2w_{x}z_{x}\nonumber \\&-\,192z^2w_{x}z_{x} +384w^3zw_{x} +384wz^3w_{x}\nonumber \\&+\,160z_{x}^3+448wzz_{x}^2+64w^4z_{x}\nonumber \\&-\,768w^2z^2z_{x}-96z^4z_{x} -256w^5z \nonumber \\ \omega _{10}= & {} +8z_{x}w_{4x}+32wzw_{4x}+40z^2z_{4x}+32wz_{x}w_{3x}\nonumber \\&-\,128w^2zw_{3x}+48z^3w_{3x}\nonumber \\&+\,224zz_{x}z_{3x} -288wz^2z_{3x} -80w_{x}z_{x}w_{xx}\nonumber \\&-\,320wzw_{x}w_{xx}-288w^2z_{x}w_{xx}\nonumber \\&-\,128w^3zw_{xx}-192wz^3w_{xx} +304z^2z_{x}w_{xx}\nonumber \\&+\,120zz_{xx}^2 +64z^2w_{x}z_{xx}\nonumber \\&+\,192z_{x}^2z_{xx} -1376wzz_{x}z_{xx}+384w^2z^2z_{xx}\nonumber \end{aligned}$$
(23)
$$\begin{aligned}&+\,48z^4z_{xx} -256ww_{x}^2z_{x}\nonumber \\&-\,256w^2zw_{x}^2 +64z^3w_{x}^2 +160zw_{x}z_{x}^2 \nonumber \\&-\,128w^3w_{x}z_{x} -192wz^2w_{x}z_{x}\nonumber \\&+\,512w^4zw_{x}-512wz_{x}^3 \nonumber \\&+\,1216w^2zz_{x}^2+272z^3z_{x}^2+256w^5z_{x} \nonumber \\ \omega _{11}= & {} -12z^2 \nonumber \\ \omega _{12}= & {} +100zz_{xx}-72z^2w_{x}\nonumber \\&+\,80z_{x}^2-96wzz_{x}+144w^2z^2+36z^4 \nonumber \\ \omega _{13}= & {} -68zz_{4x}+8z^2w_{3x}-292z_{x}z_{3x}\nonumber \\&+\,176wzz_{3x}+232zz_{x}w_{xx}+64wz^2w_{xx}\nonumber \\&-\,264z_{xx}^2 +512zw_{x}z_{xx} +816wz_{x}z_{xx}\nonumber \\&-\,896w^2zz_{xx}-\,88z^3z_{xx}-64z^2w_{x}^2\nonumber \\&+\,280w_{x}z_{x}^2 -1056wzw_{x}z_{x} +96z^4w_{x} \nonumber \\ \end{aligned}$$
$$\begin{aligned} \left( \begin{array}{l} u\\ v\\ \end{array}\right) _{t}= \left( \begin{array}{c} w_{5x}-zz_{4x}+10w_{x}w_{3x}-20w^2w_{3x}-8z^2w_{3x}-4z_{x}z_{3x}-2wzz_{3x}\\ +10w_{xx}^2 -80ww_{x}w_{xx} -32zz_{x}w_{xx}-3z_{xx}^2 -18zw_{x}z_{xx}-6wz_{x}z_{xx} \\ +16w^2zz_{xx} +8z^3z_{xx} -20w_{x}^3-18w_{x}z_{x}^2 +32wzw_{x}z_{x}+80w^4w_{x}\\ +48w^2z^2w_{x} +4z^4w_{x}+16w^2z_{x}^2+24z^2z_{x}^2 +32w^3zz_{x}+16wz^3z_{x}\\ 2zw_{4x}+2z_{x}w_{3x}-4wzw_{3x}-2z^2z_{3x}+20zw_{x}w_{xx}-4wz_{x}w_{xx}\\ -32w^2zw_{xx} -16z^3w_{xx} -8zz_{x}z_{xx}+12w_{x}^2z_{x} -64wzw_{x}^2\\ -48z^2w_{x}z_{x} +64w^3zw_{x} -32w^2w_{x}z_{x}+32wz^3w_{x}-2z_{x}^3\\ +16w^4z_{x} +48w^2z^2z_{x} +20z^4z_{x} \end{array}\right) \end{aligned}$$
(27)
$$\begin{aligned}&-\,752w^2z_{x}^2 -380z^2z_{x}^2+\,768w^3zz_{x}+96wz^3z_{x}\nonumber \\&-\,384w^4z^2-192w^2z^4-32z^6 \end{aligned}$$

The first few conserved densities of the hierarchy are listed as follows

$$\begin{aligned} \rho _{0}= & {} \alpha \nonumber \\ \rho _1= & {} 2w^2+z^2\nonumber \\ \rho _2= & {} +3ww_{4x}+10w^2w_{3x}\nonumber \\&+\,6z^2w_{3x}-20w^3w_{xx}-6wz^2w_{xx}\nonumber \\&-\,18w^2zz_{xx}-4z^3z_{xx}\nonumber \\&-\,18w^2z_{x}^2 +16w^3zz_{x} +24wz^3z_{x}\nonumber \\&+\,16w^6+24w^4z^2+12w^2z^4+2z^6\nonumber \\&\,\vdots \end{aligned}$$
(24)

These densities are sufficient to write two Magri schemes with the same Hamiltonian operators such that one of them contains the new system, and this confirms the integrability of the system (27).

2.4 Integrability of system (12)

In a manner parallel to the analysis presented earlier, and to prove the integrability of the system (12), we use the change of dependent variables

$$\begin{aligned} u\rightarrow & {} \frac{1}{2}\int (w-z){\mathrm {d}}x \end{aligned}$$
(25)
$$\begin{aligned} v\rightarrow & {} \frac{1}{2}\int (w+z){\mathrm {d}}x \end{aligned}$$
(26)

which carries the system (12) to its canonical form as

Proposition

The infinite hierarchy of system (27) can be written in not just one but two different ways

$$\begin{aligned} \left( \begin{array}{c} w_{t}\\ z_{t} \end{array} \right) = {J} \left( \begin{array}{c} \delta _{w}\\ \delta _{z} \end{array} \right) \int \rho _{1}~ \mathrm {d}x ={K} \left( \begin{array}{c} \delta _{w}\\ \delta _{z} \end{array} \right) \int \rho _{1} ~\mathrm {d}x\nonumber \\ \end{aligned}$$
(28)

with the compatible pair of Hamiltonian operators

$$\begin{aligned} {J}= \left( \begin{array}{cc} D_x &{}\quad 0 \\ 0&{} \quad 2D_x \end{array}\right) ,\quad {K^2} = \left( \begin{array}{cc} {K}_{1}^2&{}\quad {K}_{2}^2\\ {K}_{3}^2&{}\quad {K}_{4}^2 \end{array}\right) \end{aligned}$$
(29)

where

$$\begin{aligned} K_1^2= & {} D_x^7+\psi _1D_x^5+D_x^5\psi _1\nonumber \\&+\,\psi _2D_x^3+D_x^3\psi _2+\psi _3D_x\nonumber \\&+\,D_x\psi _3+8w_xD_x^{-1}w_t+8w_tD_x^{-1}w_x\nonumber \\ K_2^2= & {} D_x^6\psi _4+D_x^5\psi _5+D_x^4\psi _6+D_x^3\psi _7\nonumber \\&+\,D_x^2\psi _8+D_x\psi _9+\psi _{10}+8w_xD_x^{-1}z_t\nonumber \\&+\,8w_tD_x^{-1}z_x\nonumber \\ K_3^2= & {} -\psi _4D_x^6+\psi _5D_x^5-\psi _6D_x^4\nonumber \\&+\,\psi _7D_x^3-\psi _8D_x^2+\psi _9D_x-\psi _{10}\nonumber \\&+\,8z_xD_x^{-1}w_t +\,8z_tD_x^{-1}w_x\nonumber \\ K_4^2= & {} \psi _{11}D_x^5+D_x^5\psi _5+\psi _{12}D_x^3+D_x^3\psi _{12}\nonumber \\&+\,\psi _{13}D_x+D_x\psi _{13}+8z_xD_x^{-1}z_t\nonumber \\&+\,8z_tD_x^{-1}z_x \end{aligned}$$
(30)

where the coefficients satisfy

$$\begin{aligned} \psi _1= & {} 6w_{x}-12w^2-5z^2\nonumber \\ \psi _2= & {} -16w_{3x}+40ww_{xx}+26zz_{xx}+58w_{x}^2\nonumber \\&-\,24w^2w_{x}-12z^2w_{x}+26z_{x}^2\nonumber \\&+\,20wzz_{x} +72w^4+52w^2z^2+18z^4 \end{aligned}$$
(31)
$$\begin{aligned} \psi _3= & {} 10w_{5x}-24ww_{4x}-16zz_{4x}-100w_{x}w_{3x}\nonumber \\&+\,24w^2w_{3x}+12z^2w_{3x}-64z_{x}z_{3x}\nonumber \\&-\,12wzz_{3x} -84w_{xx}^2+64ww_{x}w_{xx}\nonumber \\&+\,28zz_{x}w_{xx}-128w^3w_{xx}-40wz^2w_{xx}\nonumber \\&-\,48z_{xx}^2-16zw_{x}z_{xx} -36wz_{x}z_{xx}\nonumber \\&-\,88w^2zz_{xx}-134z^3z_{xx} +48w_{x}^3\nonumber \\&-\,704w^2w_{x}^2-104z^2w_{x}^2-16w_{x}z_{x}^2 \nonumber \\&-\,288wzw_{x}z_{x} +24z^4w_{x}-88w^2z_{x}^2\nonumber \\&-\,216z^2z_{x}^2 -128w^3zz_{x}-24wz^3z_{x}\nonumber \\&-\,128w^6 -128w^4z^2-96w^2z^4-16z^6\nonumber \\ \psi _4= & {} -2z\nonumber \\ \psi _5= & {} 2z_{x}-4wz\nonumber \\ \psi _6= & {} -24zw_{x}+4wz_{x}+40w^2z\nonumber \\ \psi _7= & {} +36zw_{xx}+28w_{x}z_{x}-200wzw_{x}\nonumber \\&-\,40w^2z_{x}+120z^2z_{x}+80w^3z\nonumber \\ \psi _8= & {} -20zw_{3x}-8z_{x}w_{xx}+192wzw_{xx}\nonumber \\&+\,104zw_{x}^2+120ww_{x}z_{x}-144w^2zw_{x}\nonumber \\&-\,80w^3z_{x}-424zz_{x}^2-128w^4z\nonumber \\ \psi _9= & {} +4zw_{4x}+12z_{x}w_{3x}-88wzw_{3x}\nonumber \\&-\,154z^2z_{3x}-120zw_{x}w_{xx}-72wz_{x}w_{xx}\nonumber \\&+\,96w^2zw_{xx} +48z^3w_{xx} -238zz_{x}z_{xx}\nonumber \\&+\,360wz^2z_{xx}+16w_{x}^2z_{x}-128wzw_{x}^2\nonumber \\&-\,96w^2w_{x}z_{x} +312z^2w_{x}z_{x}\nonumber \\&+\,768w^3zw_{x} -96wz^3w_{x}+224z_{x}^3 +232wzz_{x}^2\nonumber \\&+\,128w^4z_{x} -672w^2z^2z_{x}-192z^4z_{x}-256w^5z\nonumber \\ \end{aligned}$$
(32)
$$\begin{aligned} \psi _{10}= & {} +8z_{x}w_{4x}+16wzw_{4x}+62z^2z_{4x}+16wz_{x}w_{3x}\nonumber \\&-\,32w^2zw_{3x}-24z^3w_{3x}\nonumber \\&-\,180wz^2z_{3x} +364zz_{x}z_{3x} \nonumber \\&+\,80w_{x}z_{x}w_{xx}+160wzw_{x}w_{xx}-296z^2z_{x}w_{xx}\nonumber \\&-\,256w^3zw_{xx}-192w^2z_{x}w_{xx} +48wz^3w_{xx}\nonumber \\&+\,186zz_{xx}^2 -176z^2w_{x}z_{xx}\nonumber \\&+\,348z_{x}^2z_{xx}-812wzz_{x}z_{xx}\nonumber \\&+\,336w^2z^2z_{xx} +96z^4z_{xx}-64ww_{x}^2z_{x}\nonumber \\&-\,512w^2zw_{x}^2 +128z^3w_{x}^2 -656zw_{x}z_{x}^2\nonumber \\&-\,256w^3w_{x}z_{x} +512w^4zw_{x}\nonumber \\&+\,912wz^2w_{x}z_{x}-272wz_{x}^3\nonumber \\&+\,1136w^2zz_{x}^2+544z^3z_{x}^2+256w^5z_{x}\nonumber \\ \psi _{11}= & {} -12z^2\nonumber \\ \psi _{12}= & {} +190zz_{xx}-144z^2w_{x}\nonumber \\&+\,74z_{x}^2-120wzz_{x}+144w^2z^2+36z^4\nonumber \\ \psi _{13}= & {} -146zz_{4x}+88z^2w_{3x}-514z_{x}z_{3x}\nonumber \\&+\,244wzz_{3x}+560zz_{x}w_{xx}-224wz^2w_{xx}\nonumber \\&-\,480z_{xx}^2 +904zw_{x}z_{xx} +1092wz_{x}z_{xx}\nonumber \\&-\,824w^2zz_{xx}-160z^3z_{xx}+512w_{x}z_{x}^2\nonumber \\&-\,520z^2w_{x}^2 -1632wzw_{x}z_{x}+864w^2z^2w_{x}\nonumber \\&+\,192z^4w_{x} -632w^2z_{x}^2-464z^2z_{x}^2\nonumber \\&+\,672w^3zz_{x} +192wz^3z_{x}\nonumber \\&-\,384w^4z^2-192w^2z^4-32z^6 \end{aligned}$$

The first few conserved densities of the system (27) are listed as follows

$$\begin{aligned} \rho _{0}= & {} \alpha \nonumber \\ \rho _1= & {} 2w^2+z^2\nonumber \\ \rho _2= & {} +3ww_{4x}-10w^2w_{3x}\nonumber \\&+\,3z^2w_{3x}-20w^3w_{xx}-24wz^2w_{xx}+18w^2zz_{xx}\nonumber \\&-\,z^3z_{xx} +18w^2z_{x}^2 +32w^3zz_{x}\nonumber \\&+\,48wz^3z_{x}+16w^6+24w^4z^2+12w^2z^4+2z^6\nonumber \\&\,\vdots \end{aligned}$$
(33)

These densities suffice to write two Magri schemes with same Hamiltonian operators that one of them contains the new system, and this in turn emphasizes the integrability of the system (27).

3 Concluding remarks

In this work, we established four fifth-order integrable coupled systems of weight 0 and 1. We examined the related recursion operator and bi-Hamiltonian formulations for the developed systems. We used the compatible pair of Hamiltonian operators to formally prove the integrability of the developed systems. The obtained results will add valuable findings to the existing integrable systems of fifth-order two-component equation. It is expected that other works will be conducted for recovering the scientific features of these systems of equations.