INTRODUCTION

The precessional motions of rigid bodies occupy a special place in the classical problem of the motion of a heavy solid and its generalizations. Applied problems related to the study of precessions of gyroscopic instruments are considered by Ishlinsky [1]. In the dynamics of a solid, precessions were studied by Grioli [2], Klein and Sommerfeld [3], the author of this article [4], and many other authors (see [5, 6]). Monograph [7] is devoted to the study of the conditions for the existence of precessional motions of a gyrostat with a variable gyrostatic moment. It provides an overview of the results obtained in this problem and formulates the main definitions of a gyrostat. The approaches adopted in the works of Wittenburg [8], Rumyantsev [9], Kharlamov [10] are of great importance in setting the problem of the motion of a gyrostat. Precessional motions are characterized by the property of constancy of the angle between two axes \({{l}_{1}},{{l}_{2}}\) passing through a fixed point, one of which (l1) is connected to the carrier body, and the other (l2) is motionless in space. In the case when one of the axes of the moving coordinate system contains the l1 axis, then it is advisable to call such a coordinate system a precessional coordinate system [11]. According to [2, 4], precession motions are divided into classes: if the speeds of precession and proper rotation are constant, then the precession is called regular; if the precession rate is constant, then the precession is called semi-regular of the first type; if only the speed of its own rotation is constant, then the precession is called semi-regular precession of the second type; in other cases, the precession is called a general type precession. The largest number of gyrostat precession movements was established for classes of regular and semi-regular precessions of the first type. It should be noted that the unique cases of precessions of a heavy rigid body described by the Euler–Poisson equations are the regular precessions obtained by Grioli [2] relative to an inclined axis and the case of Dokshevich [12], for which the product of the precession velocities and own rotation. Semi-regular precessions of the second type of a solid and a gyrostat are among the smaller number of precessions found. For example, it was proved in [4] that semiregular precessions of the second type are dynamically impossible in the classical problem. Despite this, in the problem of the motion of a gyrostat with a variable gyrostatic moment, some solutions have been obtained that have this property [7].

In this article, the conditions for the existence of semi-regular precessions of the second type of gyrostat with a variable gyrostatic moment under the action of potential and gyroscopic forces are investigated. The conditions on the parameters of the equations of motion and precession, under which the gyrostat performs a precession of the second type, are indicated.

1 STATEMENT OF THE PROBLEM

When studying the equations of motion of a gyrostat with a constant gyrostatic moment, we should take into account the well-known property of the analogy of the problem of the motion of a gyrostat under the action of potential and gyroscopic forces and the problem of the motion of a body in an ideal fluid, which was proved in a particular case by Steklov [13] and Kharlamov [14], and in the general case by Yahya [15]. For the case of a variable gyrostatic moment \({\boldsymbol{\lambda }}{\text{(}}t{\text{)}}\), there is no such analogy. Therefore, in this article we will use differential equations in the following form [6, 7, 15]:

$$A{{\dot{\boldsymbol{\omega}}}} + {{\dot{\boldsymbol{\lambda}}}}{\text{(}}t{\text{)}} = \left( {A{\boldsymbol{\omega }} + {\boldsymbol{\lambda }}{\text{(}}t{\text{)}}} \right) \cdot {\boldsymbol{\omega }} + {\boldsymbol{\omega }} \cdot B{\boldsymbol{\nu }} + {\boldsymbol{\nu }} \cdot (C{\boldsymbol{\nu }} - {\mathbf{s}}{\text{),}}$$
(1.1)
$${{\dot{\boldsymbol{\nu}}}} = {\boldsymbol{\nu }} \cdot {\boldsymbol{\omega }},$$
(1.2)

where the notation is introduced: \({\boldsymbol{\omega }} = {\text{(}}\omega _{1}^{{}}{\text{,}}\omega _{2}^{{}}{\text{,}}\omega _{3}^{{}}{\text{)}}\) is the angular velocity vector; \({\boldsymbol{\lambda }}{\text{(}}t{\text{)}} = ({{\lambda }}_{1}^{{}}{\text{(}}t{{),\lambda }}_{2}^{{}}{\text{(}}t{{),\lambda }}_{3}^{{}}{\text{(}}t{\text{)}})\) is the gyrostatic moment vector; \(A = {\text{diag(}}{{A}_{1}},{{A}_{2}},A_{3}^{{}}{\text{)}}\) is the gyrostat inertia tensor; \(B = {\text{diag(}}{{B}_{1}},{{B}_{2}},B_{3}^{{}}{\text{)}}\) is a matrix characterizing gyroscopic forces; \(C = {\text{diag(}}C_{1}^{{}},C_{2}^{{}},C_{3}^{{}}{\text{)}}\) is a matrix that determines terms that are quadratic in terms of the components of the vector \({\boldsymbol{\nu }} = {{(\nu }}_{1}^{{}}{{,\nu }}_{2}^{{}}{{,\nu }}_{3}^{{}}{\text{)}}\); \({\mathbf{s}} = {\text{(}}{{s}_{1}}{\text{,}}{{s}_{2}}{\text{,}}s_{3}^{{}}{\text{)}}\) is the vector of the generalized center of mass of the gyrostat; the dot over the variables \({\boldsymbol{\omega }}{\text{(}}t{\text{),}}\;{\boldsymbol{\lambda }}{\text{(}}t{\text{),}}\;{\boldsymbol{\nu }}{\text{(}}t{\text{)}}\) denotes differentiation with respect to time t.

Equations (1.1), (1.2) have first integrals

$${\boldsymbol{\nu }} \cdot {\boldsymbol{\nu }} = 1,\quad \left( {A{\boldsymbol{\omega }} + {\boldsymbol{\lambda }}{\text{(}}t{\text{)}}} \right) \cdot {\boldsymbol{\nu }} - \frac{1}{2}(B{\boldsymbol{\nu }} \cdot {\boldsymbol{\nu }}) = k,$$
(1.3)

where k is an arbitrary constant. All the above quantities are given in the principal moving coordinate system with unit vectors \({\mathbf{i}}_{1}^{{}},{\mathbf{i}}_{2}^{{}},{\mathbf{i}}_{3}^{{}}\).

System (1.1), (1.2) is a non-autonomous system of differential equations with respect to the variables \({\boldsymbol{\omega }}{\text{(}}t{\text{),}}\;{\boldsymbol{\lambda }}{\text{(}}t{\text{),}}\;{\boldsymbol{\nu }}{\text{(}}t{\text{)}}\). Its integration can be based on several approaches. In this article, we will assume that the rotor \({{S}_{3}}\), which carries the carrier body \({{S}_{0}}\), lies on the third coordinate axis, that is, \({\boldsymbol{\lambda }}{\text{(}}t{\text{)}} = {\text{(0,0,}}\lambda _{3}^{{}}{\text{(}}t{\text{))}}\). Then, by virtue of [10], we will consider system (1.1), (1.2) together with the equations

$${{\dot {\lambda }}}_{3}^{{}}{\text{(}}t{\text{)}} = L{\text{(}}t{\text{),}}\quad {{\lambda }}_{3}^{{}}{\text{(}}t{\text{)}} = {\text{D}}_{{\text{3}}}^{{}}[{\boldsymbol{\omega }}{\text{(}}t{\text{)}} \cdot {\mathbf{i}}_{3}^{{}} + \dot {\kappa }{\text{(}}t{\text{)}}].$$
(1.4)

here \(\dot {\kappa }{\text{(}}t{\text{)}}\) is the speed of rotation of the rotor \({{S}_{3}}\); \({\text{D}}_{{\text{3}}}^{{}}\) is the moment of inertia of the rotor \({{S}_{3}}\) relative to the axis of rotation \(Oz\); L(t) is the projection of moments and forces onto the \(Oz\)-axis from the side of the carrier body. Equations (1.4) can be studied using two approaches: if the function L(t) is given, then first the function \({{\lambda }}_{3}^{{}}{\text{(}}t{\text{)}}\) is found from the first equation of system (1.4) and equations (1.1), (1.2) are integrated, and then from the second equation (1.4) the function \(\dot {\kappa }{\text{(}}t{\text{)}}\) is defined; if \(\dot {\kappa }{\text{(}}t{\text{)}}\) is given and the function \({{\lambda }}_{3}^{{}}{\text{(}}t{\text{)}}\) is known, then the function \(L{\text{(}}t{\text{)}}\) is found from (1.4).

The problem of the motion of a gyrostat with a constant gyrostatic moment based on the Lagrange function was studied in [16].

We will study semiregular precessions using the method [17, 18]. According to this method, the gyrostat angular velocity vector can be represented as

$${\boldsymbol{\omega }} = {{\varepsilon }}({{{{\nu }}}_{3}}){\boldsymbol{\nu }} + g_{0}^{{}}{\boldsymbol{\beta }},$$
(1.5)

where \({{\varepsilon }}({{{{\nu }}}_{3}})\) is a differentiable function; \({\boldsymbol{\beta }} = ({{\beta }}_{1}^{{}},{{\beta }}_{2}^{{}},{{\beta }}_{3}^{{}})\) is a constant unit vector; \(g_{0}^{{}}\) is a constant parameter. The case \({{\varepsilon }}({{{{\nu }}}_{3}}) = {{\varepsilon }}_{0}^{{}}\), where \(\varepsilon _{0}^{{}}\) is a constant, was considered in [19].

We substitute the value (1.5) into equation (1.2):

$${{\dot{\boldsymbol{\nu}}}} = g_{0}^{{}}({\boldsymbol{\nu }} \cdot {\boldsymbol{\beta }}).$$
(1.6)

From equation (1.6) the first integral follows

$${{\beta }}_{{\text{1}}}^{{}}{{{{\nu }}}_{{\text{1}}}} + {{\beta }}_{{\text{2}}}^{{}}{{{{\nu }}}_{{\text{2}}}} + {{\beta }}_{{\text{3}}}^{{}}{{{{\nu }}}_{{\text{3}}}} = c_{0}^{{}},$$
(1.7)

which in vector form can be represented as follows: \({\boldsymbol{\beta }} \cdot {\boldsymbol{\nu }} = c_{0}^{{}}\), where \(c_{0}^{{}}\) is a constant. That is, due to the equalities \({\text{|}}{\boldsymbol{\beta }}| = 1\), \({\text{|}}{\boldsymbol{\nu }}| = 1\), the parameter \(c_{0}^{{}}\) satisfies the condition \(c_{0}^{{}} < 1\). It follows from (1.5) that the precession rate is equal to \({{\varepsilon }}({{{{\nu }}}_{3}})\), and the intrinsic rotation rate is \(g_{0}^{{}}\). That is, the precession of the gyrostat belongs to the type of semi-regular precession of the second type. We write (1.5), (1.6) in scalar form:

$$\omega _{i}^{{}} = {{\varepsilon (}}{{{{\nu }}}_{{\text{3}}}}{{)\nu }}_{i}^{{}} + g_{0}^{{}}{{\beta }}_{i}^{{}},\quad (i = \overline {1,3} ),$$
(1.8)
$${{\dot {\nu }}}_{{\text{1}}}^{{}} = g_{{\text{0}}}^{{}}({{\beta }}_{{\text{3}}}^{{}}{{{{\nu }}}_{{\text{2}}}} - {{\beta }}_{{\text{2}}}^{{}}{{{{\nu }}}_{{\text{3}}}}),\quad {{\dot {\nu }}}_{{\text{2}}}^{{}} = g_{{\text{0}}}^{{}}({{\beta }}_{{\text{1}}}^{{}}{{{{\nu }}}_{{\text{3}}}} - {{\beta }}_{{\text{3}}}^{{}}{{{{\nu }}}_{{\text{1}}}}),\quad {{\dot {\nu }}}_{{\text{3}}}^{{}} = g_{{\text{0}}}^{{}}({{\beta }}_{{\text{2}}}^{{}}{{{{\nu }}}_{{\text{1}}}} - {{\beta }}_{{\text{1}}}^{{}}{{{{\nu }}}_{{\text{2}}}}).$$
(1.9)

Using the equalities \({{\nu }}_{{\text{1}}}^{{\text{2}}} + {{\nu }}_{{\text{2}}}^{{\text{2}}} + {{\nu }}_{{\text{3}}}^{{\text{2}}} = 1\) and (1.7), we find the functions \({{{{\nu }}}_{{\text{1}}}}{\text{(}}{{{{\nu }}}_{3}})\), \({{{{\nu }}}_{{\text{2}}}}{\text{(}}{{{{\nu }}}_{3}})\):

$$\begin{gathered} {{\nu }}_{{\text{1}}}^{{}}({{\nu }}_{{\text{3}}}^{{}}) = \frac{{\text{1}}}{{\kappa _{{\text{0}}}^{{\text{2}}}}}[{{\beta }}_{{\text{1}}}^{{}}{\text{(}}c_{{\text{0}}}^{{}} - {{\beta }}_{{\text{3}}}^{{}}{{{{\nu }}}_{{\text{3}}}}{\text{)}} + {{\beta }}_{{\text{2}}}^{{}}\sqrt {F({{\nu }}_{{\text{3}}}^{{}})} ], \\ {{\nu }}_{{\text{2}}}^{{}}({{\nu }}_{{\text{3}}}^{{}}) = \frac{{\text{1}}}{{\kappa _{{\text{0}}}^{{\text{2}}}}}[{{\beta }}_{{\text{2}}}^{{}}{\text{(}}c_{{\text{0}}}^{{}} - {{\beta }}_{{\text{3}}}^{{}}{{{{\nu }}}_{{\text{3}}}}{\text{)}} - {{\beta }}_{{\text{1}}}^{{}}\sqrt {F({{\nu }}_{{\text{3}}}^{{}})} ], \\ \end{gathered} $$
(1.10)

where \(\kappa _{0}^{2} = {{\beta }}_{{\text{1}}}^{{\text{2}}} + {{\beta }}_{{\text{2}}}^{{\text{2}}}\), and the function \(F({{\nu }}_{3}^{{}})\) is

$$F({{\nu }}_{{\text{3}}}^{{}}) = - {{\nu }}_{3}^{2} + 2c_{0}^{{}}{{\beta }}_{{\text{3}}}^{{}}{{{{\nu }}}_{3}} + (\kappa _{0}^{2} - c_{0}^{2}).$$
(1.11)

Substituting \({{{{\nu }}}_{{\text{1}}}}{\text{(}}{{{{\nu }}}_{3}})\), \({{{{\nu }}}_{{\text{2}}}}{\text{(}}{{{{\nu }}}_{3}})\) from (1.10) into the third equation of system (1.9), we obtain that the function \({{{{\nu }}}_{3}}(t)\) can be obtained by inverting the integral [17]

$$\int\limits_{{{\nu }}_{{\text{3}}}^{{{\text{(0)}}}}}^{{{\nu }}_{{\text{3}}}^{{}}} {\frac{{d{{\nu }}_{{\text{3}}}^{{}}}}{{\sqrt {F({{\nu }}_{{\text{3}}}^{{}})} }}} = g_{0}^{{}}(t - t_{0}^{{}}).$$
(1.12)

Then the functions \({{\nu }}_{i}^{{}}{\text{(}}t{\text{)}}\) \({\text{(}}i = 1,2{\text{)}}\), \({{\omega }}_{i}^{{}}{\text{(}}t{\text{)}}\) \({\text{(}}i = \overline {1,3} {\text{)}}\) are found from equations (1.10), (1.8). Since the function (1.11) satisfies the condition \(F({{\nu }}_{3}^{{}}) < 0\) for \({\text{|}}{{{{\nu }}}_{3}}{\text{|}} > 1\), then for a real value of the parameter \({{\mu }}_{0}^{{}} = \sqrt {1 - c_{0}^{2}} \) the roots of the equation \(F({{\nu }}_{3}^{{}}) = 0\) are real. That is, from (1.11), (1.12) we find

$$({{\nu }}_{{\text{3}}}^{{}})_{{{\text{1,2}}}}^{{}} = c_{{\text{0}}}^{{}}{{\beta }}_{{\text{3}}}^{{}} \pm {{\mu }}_{0}^{{}}\kappa _{0}^{{}}{{\sin\psi ,}}$$
(1.13)

where, by virtue of the third equation from (1.9), \({{\psi }} = g_{0}^{{}}t\). Choosing the plus sign in (1.13) for definiteness, from (1.13), (1.10) we obtain

$$\begin{gathered} \nu _{1}^{{}}{\text{ }}({{\psi }}) = h_{0}^{{}} + h_{1}^{{}}{{\cos\psi }} + h_{2}^{{}}{{\sin\psi }},\quad \nu _{2}^{{}}({{\psi }}) = r_{0}^{{}} + r_{1}^{{}}{{\cos\psi }} + r_{2}^{{}}{{\sin\psi }}, \\ \nu _{3}^{{}}({{\psi }}) = a_{0}^{{}} + a_{2}^{{}}{{\sin\psi }}{\text{.}} \\ \end{gathered} $$
(1.14)

Here we introduced the notation

$$\begin{gathered} h_{0}^{{}} = c_{0}^{{}}{{\beta }}_{1}^{{}},\quad h_{1}^{{}} = \frac{{{{\beta }}_{2}^{{}}{{\mu }}_{0}^{{}}}}{{\kappa _{0}^{{}}}},\quad h_{2}^{{}} = - \frac{{{{\beta }}_{1}^{{}}{{\beta }}_{3}^{{}}{{\mu }}_{0}^{{}}}}{{\kappa _{0}^{{}}}}, \\ r_{0}^{{}} = c_{0}^{{}}{{\beta }}_{2}^{{}},\quad r_{1}^{{}} = - \frac{{{{\beta }}_{1}^{{}}{{\mu }}_{0}^{{}}}}{{\kappa _{0}^{{}}}},\quad r_{2}^{{}} = - \frac{{{{\beta }}_{2}^{{}}{{\beta }}_{3}^{{}}{{\mu }}_{0}^{{}}}}{{\kappa _{0}^{{}}}}{\text{,}}\quad a_{0}^{{}} = c_{0}^{{}}{{\beta }}_{3}^{{}},\quad a_{2}^{{}} = \kappa _{0}^{{}}{{\mu }}_{0}^{{}}. \\ \end{gathered} $$
(1.15)

Let us note the form of solution (1.14) and notation (1.15) in the case of \({{\beta }}_{3}^{{}} = 0\), which was considered in [19] when studying regular gyrostat precessions (\({{\varepsilon }}({{{{\nu }}}_{3}}) = {{\varepsilon }}_{0}^{{}}\)):

$$\begin{gathered} \nu _{1}^{{}}({{\psi }}) = h_{0}^{{}} + h_{1}^{{}}{{\cos\psi ,}}\quad \nu _{2}^{{}}({{\psi }}){\text{ }} = r_{0}^{{}} + r_{1}^{{}}{{\cos\psi }},\quad \nu _{3}^{{}}{\text{ }}({{\psi }}) = a_{2}^{{}}{{\sin\psi ,}} \\ h_{0}^{{}} = c{{\beta }}_{1}^{{}},\quad h_{1}^{{}} = {{\beta }}_{{\text{2}}}^{{}}{{\mu }}_{0}^{{}},\quad h_{2}^{{}} = 0,\quad r_{0}^{{}} = c_{0}^{{}}{{\beta }}_{2}^{{}}, \\ r_{1}^{{}} = - {{\beta }}_{{\text{1}}}^{{}}{{\mu }}_{0}^{{}},\quad r_{2}^{{}} = 0,\quad a_{0}^{{}} = 0,\quad a_{2}^{{}} = \mu _{0}^{{}}. \\ \end{gathered} $$
(1.16)

Let us pose the problem: to determine the conditions for the existence of a solution (1.8), (1.14) of equation (1.1).

2 STUDY OF EQUATION (1.1)

Let us write equation (1.1) in scalar form:

$$A_{1}^{{}}\dot {\omega }_{1}^{{}}{\text{ }} = (A_{2}^{{}} - A_{3}^{{}})\omega _{2}^{{}}\omega _{3}^{{}} - {{\lambda }_{3}}(t)\omega _{2}^{{}} + \omega _{2}^{{}}B_{3}^{{}}{{\nu }_{3}} - \omega _{3}^{{}}B_{2}^{{}}{{\nu }_{2}} + s_{2}^{{}}{{\nu }_{3}} - s_{3}^{{}}{{\nu }_{2}} + (C_{3}^{{}} - C_{2}^{{}})\nu _{2}^{{}}{{\nu }_{3}},$$
(2.1)
$$A_{2}^{{}}\dot {\omega }_{2}^{{}}{\text{ }} = (A_{3}^{{}} - A_{1}^{{}})\omega _{3}^{{}}\omega _{1}^{{}} + {{\lambda }_{3}}(t)\omega _{1}^{{}} + \omega _{3}^{{}}B_{1}^{{}}{{\nu }_{1}} - \omega _{1}^{{}}B_{3}^{{}}{{\nu }_{3}} + s_{3}^{{}}{{\nu }_{1}} - s_{1}^{{}}{{\nu }_{3}} + (C_{1}^{{}} - C_{3}^{{}})\nu _{3}^{{}}{{\nu }_{1}},$$
(2.2)
$${{\dot {\lambda }}_{3}}(t) + A_{3}^{{}}\dot {\omega }_{3}^{{}}{\text{ }} = (A_{1}^{{}} - A_{2}^{{}})\omega _{1}^{{}}\omega _{2}^{{}} + \omega _{1}^{{}}B_{2}^{{}}{{\nu }_{2}} - \omega _{2}^{{}}B_{1}^{{}}{{\nu }_{1}} + s_{1}^{{}}{{\nu }_{2}} - s_{2}^{{}}{{\nu }_{1}} + (C_{2}^{{}} - C_{1}^{{}})\nu _{1}^{{}}{{\nu }_{2}}.$$
(2.3)

Let us substitute expressions \(\omega _{i}^{{}}\) from (1.8) into (2.1)–(2.3) and use equations (1.9). Then we get a system of three differential equations

$$\begin{gathered} \lambda _{3}^{{}}{\text{(}}t{\text{)}}({v}_{2}^{{}}\varepsilon ({{\nu }_{3}}) + \beta _{2}^{{}}g_{0}^{{}}) = A_{1}^{{}}g_{0}^{{}}{v}_{1}^{{}}\varepsilon {\kern 1pt} '({{\nu }_{3}})(\beta _{1}^{{}}{v}_{2}^{{}} - \beta _{2}^{{}}{v}_{1}^{{}}) + A_{1}^{{}}g_{0}^{{}}\varepsilon ({{\nu }_{3}})(\beta _{2}^{{}}{v}_{3}^{{}} - \beta _{3}^{{}}{v}_{2}^{{}}) \\ \; + \nu _{2}^{{}}{{\nu }_{3}}[\varepsilon _{{}}^{2}({{\nu }_{3}})(A_{2}^{{}} - A_{3}^{{}}) + \varepsilon ({{\nu }_{3}})(B_{3}^{{}} - B_{2}^{{}}) + C_{3}^{{}} - C_{2}^{{}}] \\ \; + \nu _{2}^{{}}\{ \beta _{3}^{{}}g_{0}^{{}}[\varepsilon ({{\nu }_{3}})(A_{2}^{{}} - A_{3}^{{}}) - B_{2}^{{}}] - s_{3}^{{}}\} \\ \; + \nu _{3}^{{}}\{ \beta _{2}^{{}}g_{0}^{{}}[\varepsilon ({{\nu }_{3}})(A_{2}^{{}} - A_{3}^{{}}) + B_{3}^{{}}] + s_{2}^{{}}\} + \beta _{2}^{{}}\beta _{3}^{{}}g_{0}^{2}(A_{2}^{{}} - A_{3}^{{}}) \\ \end{gathered} $$
(2.4)
$$\begin{gathered} - \lambda _{3}^{{}}{\text{(}}t{\text{)}}({v}_{1}^{{}}\varepsilon ({{\nu }_{3}}) + \beta _{1}^{{}}g_{0}^{{}}) = A_{2}^{{}}g_{0}^{{}}{v}_{2}^{{}}\varepsilon {\kern 1pt} '({{\nu }_{3}})(\beta _{1}^{{}}{v}_{2}^{{}} - \beta _{2}^{{}}{v}_{1}^{{}}) + A_{2}^{{}}g_{0}^{{}}\varepsilon ({{\nu }_{3}})(\beta _{3}^{{}}{v}_{1}^{{}} - \beta _{1}^{{}}{v}_{3}^{{}}) \\ \; + \nu _{1}^{{}}{{\nu }_{3}}[\varepsilon _{{}}^{2}({{\nu }_{3}})(A_{3}^{{}} - A_{1}^{{}}) + \varepsilon ({{\nu }_{3}})(B_{1}^{{}} - B_{3}^{{}}) + C_{1}^{{}} - C_{3}^{{}}] \\ \; + \nu _{1}^{{}}\{ \beta _{3}^{{}}g_{0}^{{}}[\varepsilon ({{\nu }_{3}})(A_{3}^{{}} - A_{1}^{{}}) + B_{1}^{{}}] + s_{3}^{{}}\} \\ \; + \nu _{3}^{{}}\{ \beta _{1}^{{}}g_{0}^{{}}[\varepsilon ({{\nu }_{3}})(A_{3}^{{}} - A_{1}^{{}}) - B_{3}^{{}}] - s_{1}^{{}}\} + \beta _{1}^{{}}\beta _{3}^{{}}g_{0}^{2}(A_{3}^{{}} - A_{1}^{{}}), \\ \end{gathered} $$
(2.5)
$$\begin{gathered} \dot {\lambda }_{3}^{{}}{\text{(}}t{\text{)}} = A_{3}^{{}}{v}_{3}^{{}}\varepsilon {\kern 1pt} '({{\nu }_{3}})(\beta _{1}^{{}}{v}_{2}^{{}} - \beta _{2}^{{}}{v}_{1}^{{}}) + A_{3}^{{}}g_{0}^{{}}\varepsilon ({{\nu }_{3}})(\beta _{1}^{{}}{v}_{2}^{{}} - \beta _{2}^{{}}{v}_{1}^{{}}) \\ \, + \nu _{1}^{{}}{{\nu }_{2}}[\varepsilon _{{}}^{2}({{\nu }_{3}})(A_{1}^{{}} - A_{2}^{{}}) + \varepsilon ({{\nu }_{3}})(B_{2}^{{}} - B_{1}^{{}}) + C_{2}^{{}} - C_{1}^{{}}] \\ \, + \nu _{1}^{{}}\{ \beta _{2}^{{}}g_{0}^{{}}[\varepsilon ({{\nu }_{3}})(A_{1}^{{}} - A_{2}^{{}}) - B_{1}^{{}}] - s_{2}^{{}}\} \\ \, + \nu _{2}^{{}}\{ \beta _{1}^{{}}g_{0}^{{}}[\varepsilon ({{\nu }_{3}})(A_{1}^{{}} - A_{2}^{{}}) + B_{2}^{{}}] + s_{1}^{{}}\} + \beta _{1}^{{}}\beta _{2}^{{}}g_{0}^{2}(A_{1}^{{}} - A_{2}^{{}}). \\ \end{gathered} $$
(2.6)

The choice of the form of differential equations (2.4)–(2.6) is related to the applied technique for studying them in further transformations.

In some cases, it is advisable to use equations (2.1), (2.2), excluding the function \(\lambda _{3}^{{}}{\text{(}}t{\text{)}}\) from them:

$$\begin{gathered} \frac{1}{2}{{(A_{1}^{{}}\omega _{1}^{2} + A_{2}^{{}}\omega _{2}^{2})}^{ \bullet }} = (A_{2}^{{}} - A_{1}^{{}})\omega _{1}^{{}}\omega _{2}^{{}}\omega _{3}^{{}} + \omega _{3}^{{}}(B_{1}^{{}}{{\nu }_{1}}\omega _{2}^{{}} - B_{2}^{{}}{{\nu }_{2}}\omega _{1}^{{}}) \\ \; + s_{3}^{{}}(\omega _{2}^{{}}{{\nu }_{1}} - \omega _{1}^{{}}{{\nu }_{2}}) + {{\nu }_{3}}(\omega _{1}^{{}}s_{2}^{{}} - \omega _{2}^{{}}s_{1}^{{}}) + {{\nu }_{3}}[{{\nu }_{2}}\omega _{1}^{{}}(C_{3}^{{}} - C_{2}^{{}}) - {{\nu }_{1}}\omega _{2}^{{}}(C_{3}^{{}} - C_{1}^{{}})]. \\ \end{gathered} $$
(2.7)

In the general case, substituting functions (1.14) into equations (2.4), (2.5) and excluding the function \(\lambda _{3}^{{}}{\text{(}}t{\text{)}}\) from the obtained equations, we arrive at a Riccati type equation, the solution of which is not possible to establish. Therefore, in further transformations, we assume that following conditions hold:

$$A_{2}^{{}} = A_{1}^{{}},\quad B_{2}^{{}} = B_{1}^{{}},\quad C_{2}^{{}} = C_{1}^{{}}.$$
(2.8)

In addition, we will consider two independent options for additional restrictions on parameters:

$$1.\,\,s_{3}^{{}} = 0,\quad s_{1}^{{}} = d_{0}^{{}}\beta _{1}^{{}},\quad s_{2}^{{}} = d_{0}^{{}}\beta _{2}^{{}}$$
(2.9)
$$2.\,\,s_{1}^{{}} = 0,\quad s_{2}^{{}} = 0,\quad s_{3}^{{}} \ne 0,$$
(2.10)

where d0 is a parameter. In case (2.9), we transform equation (2.7) on the basis of (1.8), (1.9) to the form

$$\frac{1}{2}A_{1}^{{}}(\omega _{1}^{2} + \omega _{2}^{2})_{{{{\nu }_{3}}}}^{'} = \frac{{d_{0}^{{}} + g_{0}^{{}}B_{1}^{{}}}}{{g_{0}^{{}}}}\nu _{3}^{{}}\varepsilon ({{\nu }_{3}}) + B_{1}^{{}}g_{0}^{{}}\beta _{3}^{{}} + (C_{1}^{{}} - C_{3}^{{}}){{\nu }_{3}}.$$
(2.11)

By virtue of (2.8), (2.9), these equalities can be interpreted as generalized S.V. Kovalevskaya conditions.

When conditions (2.10) are satisfied, we represent equation (2.7) as follows:

$$\frac{1}{2}A_{1}^{{}}(\omega _{1}^{2} + \omega _{2}^{2})_{{{{\nu }_{3}}}}^{'} = B_{1}^{{}}\nu _{3}^{{}}\varepsilon ({{\nu }_{3}}) + (s_{3}^{{}} + \beta _{3}^{{}}g_{0}^{{}}B_{1}^{{}}) + (C_{1}^{{}} - C_{3}^{{}}){{\nu }_{3}}.$$
(2.12)

Analogues of the first integrals following from (2.12) were considered in [20]. We write equation (2.3) in case (2.9):

$$A_{3}^{{}}\omega _{3}^{{}}({{\nu }_{3}}) + {{\lambda }_{3}}({{\nu }_{3}}) = - \frac{{d_{0}^{{}} + B_{1}^{{}}g_{0}^{{}}}}{{g_{0}^{{}}}}\nu _{3}^{{}} + B_{0}^{{}},$$
(2.13)

where \(B_{0}^{{}}\) is an arbitrary constant. If we take into account conditions (2.10) in equation (2.3) and the third equation from (1.9), then we obtain

$$A_{3}^{{}}\omega _{3}^{{}}({{\nu }_{3}}) + {{\lambda }_{3}}({{\nu }_{3}}) = - B_{1}^{{}}\nu _{3}^{{}} + l_{0}^{{}}.$$
(2.14)

Here, l0 is an arbitrary constant. The analogy of equations (2.11) and (2.12), as well as (2.13), (2.14) is obvious.

Let us consider a linear combination of equations (2.4), (2.5), multiplying equation (2.4) by \({{\nu }_{1}}\), equation (2.5) by \({{\nu }_{2}}\), and adding the left and right parts of the resulting equations. Then, by virtue of the equation \({\dot {v}}_{3}^{{}} = g_{0}^{{}}(\beta _{2}^{{}}{{\nu }_{1}} - \beta _{1}^{{}}{{\nu }_{2}})\), we find

$$\begin{gathered} \{ A_{1}^{{}}[(\nu _{3}^{2} - 1)\varepsilon ({{\nu }_{3}})]_{{}}^{'} - A_{3}^{{}}\nu _{3}^{{}}\varepsilon ({{\nu }_{3}}) + B_{3}^{{}}\nu _{3}^{{}} + g_{0}^{{}}\beta _{3}^{{}}(A_{1}^{{}} - A_{3}^{{}}) - {{\lambda }_{3}}({{\nu }_{3}})\} {{{\dot {\nu }}}_{3}} \\ \, + s_{2}^{{}}{{\nu }_{1}} - s_{1}^{{}}{{\nu }_{2}} = 0. \\ \end{gathered} $$
(2.15)

In case (2.9), (2.13), it follows from (2.15)

$$\varepsilon ({{\nu }_{3}}) = \frac{1}{{1 - \nu _{3}^{2}}}(E_{2}^{{}}\nu _{3}^{2} + E_{1}^{{}}\nu _{3}^{{}} + E_{0}^{{}}),$$
(2.16)

where E0 is an arbitrary constant and \({{E}_{2}},{{E}_{1}}\) are:

$$E_{2}^{{}} = \frac{1}{{2g_{0}^{{}}A_{1}^{{}}}}[g_{0}^{{}}(B_{1}^{{}} + B_{3}^{{}}) + 2d_{0}^{{}}],\quad E_{1}^{{}} = \frac{1}{{A_{1}^{{}}}}(\beta _{3}^{{}}g_{0}^{{}}A_{1}^{{}} - B_{0}^{{}}).$$
(2.17)

Let us write equation (2.15) under conditions (2.10), (2.14):

$$\varepsilon ({{\nu }_{3}}) = \frac{1}{{1 - \nu _{3}^{2}}}(G_{2}^{{}}\nu _{3}^{2} + G_{1}^{{}}\nu _{3}^{{}} + G_{0}^{{}}),$$
(2.18)

where G0 is an arbitrary constant and \(G_{2}^{{}},G_{1}^{{}}\) have the form

$$G_{2}^{{}} = \frac{1}{{2A_{1}^{{}}}}(B_{1}^{{}} + B_{3}^{{}}),\quad G_{1}^{{}} = \frac{{\beta _{3}^{{}}g_{0}^{{}}A_{1}^{{}} - l_{0}^{{}}}}{{A_{1}^{{}}}}.$$
(2.19)

It follows from (2.17) and (2.19) that the quantity \(E_{2}^{{}}\) at \(d_{0}^{{}} = 0\) will take the value \(G_{2}^{{}}\), and to obtain the value \(G_{1}^{{}}\) from (2.17), we must assume \(B_{0}^{{}} = l_{0}^{{}}\). However, equations (2.11), (2.12) do not have such an analogy.

3 CASE (2.9)

At the first stage, we will study this case under the condition

$$d_{0}^{{}} + g_{0}^{{}}B_{1}^{{}} = 0.$$
(3.1)

When equality (3.1) is satisfied, we write equation (2.11) in the form of its first integral

$$\begin{gathered} \frac{{A_{1}^{{}}}}{2}[({\text{1}} - {v}_{3}^{2})\varepsilon _{{}}^{2}({{\nu }_{3}}) + 2g_{0}^{{}}\varepsilon ({{\nu }_{3}})(c_{0}^{{}} - \beta _{3}^{{}}{{\nu }_{3}} - g_{0}^{{}}B_{1}^{{}}\beta _{3}^{{}}{{\nu }_{3}}) \\ \, - \beta _{3}^{{}}g_{0}^{{}}B_{1}^{{}}{{\nu }_{3}} + \frac{1}{2}(C_{3}^{{}} - C_{1}^{{}}){v}_{3}^{2} = D_{{\text{0}}}^{{}}, \\ \end{gathered} $$
(3.2)

where D0 is an arbitrary constant. In equation (3.2), we take into account the invariant relation (1.8). Let us substitute the value \(\varepsilon ({{\nu }_{3}})\) from equation (2.16) into equality (3.2) and require that the resulting equality be an identity with respect to the variable \({{\nu }_{3}}\). Then we obtain the following algebraic system for the parameters of the problem:

$$C_{1}^{{}} - C_{{\text{3}}}^{{}} + A_{1}^{{}}E_{2}^{2} = 0,$$
(3.3)
$$B_{0}^{{}} = \frac{{g_{0}^{{}}{{\beta }_{3}}B_{1}^{{}}}}{{E_{2}^{{}}}},\quad E_{1}^{2} + (E_{0}^{{}} + E_{2}^{{}})_{{}}^{2} + 2g_{0}^{{}}c_{0}^{{}}(E_{0}^{{}} + E_{2}^{{}}) + E_{1}^{{}}(E_{1}^{{}} - 2g_{0}^{{}}\beta _{3}^{{}}) = 0,$$
(3.4)
$$E_{0}^{{}}(E_{1}^{{}} - \beta _{3}^{{}}g_{0}^{{}}) + 2g_{0}^{{}}c_{0}^{{}}E_{1}^{{}} - \frac{{\beta _{3}^{{}}g_{0}^{{}}B_{1}^{{}} - l_{0}^{{}}}}{{A_{1}^{{}}}} = 0,$$
(3.5)
$$D_{{\text{0}}}^{{}} = \frac{{A_{1}^{{}}E_{0}^{{}}}}{2}(E_{0}^{{}} + 2g_{0}^{{}}c_{0}^{{}}).$$
(3.6)

We shall considered equation (3.3) as a condition on the parameters \({{C}_{1}},{{C}_{3}}\). The first equality from (3.4), due to (2.17) and assumption (3.1), for which the value of q has the form

$$E_{2}^{{}} = \frac{{B_{3}^{{}} - B_{1}^{{}}}}{{2A_{1}^{{}}}}.$$
(3.7)

let’s write it like this

$$B_{0}^{{}} = \frac{{2{{\beta }_{3}}g_{0}^{{}}A_{1}^{{}}B_{1}^{{}}}}{{B_{3}^{{}} - B_{1}^{{}}}}.$$
(3.8)

Based on the value (3.8), we express the parameter E1 from (2.17) in terms of the parameters of the problem:

$$E_{1}^{{}} = \frac{{{{\beta }_{3}}g_{0}^{{}}(B_{3}^{{}} - 3B_{1}^{{}})}}{{B_{3}^{{}} - B_{1}^{{}}}}.$$
(3.9)

From the second equation of system (3.4) and equation (3.6), we find the values of \(c_{0}^{{}},E_{0}^{{}}\)

$$c_{0}^{{}} = \delta _{0}^{{}}\frac{{2{{\beta }_{3}}B_{1}^{{}}}}{{B_{3}^{{}} - B_{1}^{{}}}},\quad E_{0}^{{}} = \delta _{0}^{{}}E_{1}^{{}} - \frac{{B_{3}^{{}} - B_{1}^{{}}}}{{2A_{1}^{{}}}},$$
(3.10)

where \(\delta _{0}^{{}} = \pm 1\). Equation (3.6) on the basis of (3.10) serves to determine the constant \(D_{0}^{{}}\). Let us study the function (2.16) taking into account (3.7), (3.10). For definiteness, we set \(\delta _{0}^{{}} = 1\). Then

$$\varepsilon ({{\nu }_{3}}) = \frac{1}{{1 - \nu _{3}^{{}}}}(E_{2}^{{}}\nu _{3}^{{}} + E_{2}^{*}),\quad E_{2}^{*} = \frac{{2{{\beta }_{3}}g_{0}^{{}}(B_{3}^{{}} - 3B_{1}^{{}}) - (B_{3}^{{}} - B_{1}^{{}})_{{}}^{2}}}{{2A_{1}^{{}}(B_{3}^{{}} - B_{1}^{{}})}}.$$
(3.11)

In order for the function \(\varepsilon ({{\nu }_{3}})\) not to take a constant value, we assume that the conditions \({{\beta }_{3}} \ne 0\), \(B_{3}^{{}} - 3B_{1}^{{}} \ne 0\) are satisfied. Thus, the function \(\varepsilon ({{\nu }_{3}})\) from (3.11) is a linear-fractional function of ν3 = \(a_{0}^{{}} + a_{2}^{{}}{\text{sin}}g_{0}^{{}}t\). Let us indicate the value of \({{\lambda }_{3}}({{\nu }_{3}})\) in the case under consideration. From equations (2.13), (3.11) we have

$${{\lambda }_{3}}({{\nu }_{3}}) = \frac{{{{\beta }_{3}}g_{0}^{{}}[2A_{1}^{{}}B_{1}^{{}} - A_{3}^{{}}(B_{3}^{{}} - B_{1}^{{}})]}}{{B_{3}^{{}} - B_{1}^{{}}}} - \frac{{A_{3}^{{}}\nu _{3}^{{}}(E_{2}^{{}}\nu _{3}^{{}} + E_{2}^{*})}}{{1 - \nu _{3}^{{}}}}.$$
(3.12)

For the final solution of the problem of the conditions for the existence of semi-regular gyrostat precessions, it is necessary to consider equations (1.4) together with the value of function (3.12), setting \({{\nu }_{3}}(t) = a_{0}^{{}} + a_{2}^{{}}{\text{sin}}g_{0}^{{}}t\). Due to the obviousness of these transformations, we confine ourselves to their explanations.

4 CASE \(d_{{\mathbf{0}}}^{{}} + g_{{\mathbf{0}}}^{{}}B_{{\mathbf{1}}}^{{}} \ne {\mathbf{0}}\)

Let us consider equation (2.11) under the condition \(d_{0}^{{}} + g_{0}^{{}}B_{1}^{{}} \ne 0\). The function \(\varepsilon ({{\nu }_{3}})\) has the form (2.16) with the notation (2.17). Let us substitute \(\omega _{1}^{{}}\), \(\omega _{2}^{{}}\) from (1.8) into it and take equations (1.9) into account. Then we get a differential equation for the function \(\varepsilon ({{\nu }_{3}})\):

$$\begin{gathered} A_{1}^{{}}\varepsilon _{{}}^{'}({{\nu }_{3}})[({\text{1}} - {v}_{3}^{2})\varepsilon ({{\nu }_{3}}) + g_{0}^{{}}(c_{0}^{{}} - \beta _{3}^{{}}){v}_{3}^{{}}] - A_{1}^{{}}{{\nu }_{3}}\varepsilon _{{}}^{2}({{\nu }_{3}}) \\ \; - \frac{1}{{g_{0}^{{}}}}\varepsilon ({{\nu }_{3}})[(d_{0}^{{}} + B_{1}^{{}}g_{0}^{{}}){{\nu }_{3}} + g_{0}^{2}\beta _{3}^{{}}A_{1}^{{}}] + g_{0}^{{}}(C_{3}^{{}} - C_{1}^{{}}){v}_{3}^{{}} - {{\beta }_{3}}g_{0}^{{}}B_{1}^{{}} = 0. \\ \end{gathered} $$
(4.1)

We introduce the function (2.16) into equation (4.1). It is convenient to present the result as follows:

$$\begin{gathered} A_{1}^{{}}[E_{1}^{{}}\nu _{3}^{2} + 2(E_{0}^{{}} + E_{2}^{{}}){{\nu }_{3}} + E_{1}^{{}}](E_{2}^{{}}\nu _{3}^{2} + \tilde {E}_{1}^{{}}{{\nu }_{3}} + \tilde {E}_{0}^{{}}) \\ \, - A_{1}^{{}}{{\nu }_{3}}(E_{2}^{{}}\nu _{3}^{2} + E_{1}^{{}}{{\nu }_{3}} + E_{0}^{{}})_{{}}^{2} - (1 - {v}_{3}^{2})(E_{2}^{{}}{v}_{3}^{2} + E_{1}^{{}}{{\nu }_{3}} + E_{0}^{{}})(R_{0}^{{}} + R_{1}^{{}}{{\nu }_{3}}) \\ \, + (1 - {v}_{3}^{2})_{{}}^{2}[(C_{3}^{{}} - C_{1}^{{}}){v}_{3}^{{}} - {{\beta }_{3}}B_{1}^{{}}g_{0}^{{}}] = 0, \\ \end{gathered} $$
(4.2)

where

$$\tilde {E}_{0}^{{}} = E_{0}^{{}} + c_{0}^{{}}g_{0}^{{}},\quad \tilde {E}_{1}^{{}} = E_{1}^{{}} - {{\beta }_{3}}g_{0}^{{}},\quad R_{1}^{{}} = \frac{{d_{0}^{{}} + g_{0}^{{}}B_{1}^{{}}}}{{g_{0}^{{}}}},\quad R_{0}^{{}} = {{\beta }_{3}}g_{0}^{{}}A_{1}^{{}}.$$
(4.3)

Let us require that equation (4.2) be an identity in the variable \({{\nu }_{3}}\). The zero coefficient of \(\nu _{3}^{5}\) has the form

$$E_{2}^{{}}R_{1}^{{}} + C_{3}^{{}} - C_{1}^{{}} - A_{1}^{{}}E_{2}^{2} = 0.$$
(4.4)

To simplify other conditions, we express the quantity \(C_{3}^{{}} - C_{1}^{{}}\) from equality (4.4) and substitute it into (4.2). Let us write down the equality to zero of the coefficient at \(\nu _{3}^{3}\). Using equations (4.3), we obtain

$$R_{1}^{{}}(E_{0}^{{}} + E_{2}^{{}}) = 0.$$
(4.5)

Let us show that the equality \(R_{1}^{{}} = 0\) must hold. Suppose the contrary; then from (4.5) we have

$$E_{0}^{{}} + E_{2}^{{}} = 0.$$
(4.6)

Condition (4.6) allows us to consider equation (4.2) for \({v}_{3}^{{}} = \pm 1\):

$$E_{1}^{{}}(2c_{0}^{{}}g_{0}^{{}} \pm E_{1}^{{}} \mp 2g_{0}^{{}}{{\beta }_{3}}) = 0.$$
(4.7)

The case \(E_{1}^{{}} = 0\), due to (4.6) for the function \(\varepsilon ({{\nu }_{3}})\) from (2.16), leads to a constant value \(\varepsilon ({{\nu }_{3}})\). Therefore, in (4.7) it is necessary to put

$$c_{0}^{{}} = 0,\quad E_{1}^{{}} = 2g_{0}^{{}}{{\beta }_{3}}.$$
(4.8)

From the equalities to zero in equation (4.2) of the coefficients at \({v}_{3}^{4}\) and the free term, it follows

$$E_{1}^{{}}R_{1}^{{}} = E_{2}^{{}}R_{0}^{{}} + g_{0}^{{}}B_{1}^{{}}{{\beta }_{3}},\quad R_{0}^{{}}E_{0}^{{}} - g_{0}^{{}}B_{1}^{{}}{{\beta }_{3}} = 0.$$
(4.9)

Since by assumption (4.6) \(E_{0}^{{}} = - E_{2}^{{}}\), then by virtue of \(E_{1}^{{}} \ne 0\) from (4.9) we obtain \(R_{1}^{{}} = 0\), which was to be proved. So, the option when the parameters satisfy the condition \(d_{0}^{{}} + g_{0}^{{}}B_{1}^{{}} \ne 0\) is impossible.

5 CASE (2.10)

Let us consider equations (2.11), (2.12) and functions (2.13), (2.14), (2.16), (2.18). In order to study this option, we can formally go from equation (2.11) to equation (2.12), from equation (2.13) go to equation (2.14), from function (2.16) to function (2.18), then case (2.10) is obtained from case (2.9) by setting \(d_{0}^{{}} = 0\) \(B_{0}^{{}} = l_{0}^{{}}\) in it and replacing the expression \(B_{1}^{{}}g_{0}^{{}}{{\beta }_{3}}\) with the expression \(s_{3}^{{}} + B_{1}^{{}}g_{0}^{{}}{{\beta }_{3}}\). Therefore, equation (4.1) will correspond to an equation in which \(R_{1}^{{}} = B_{1}^{{}}\), and instead of the last term in (4.1) it is necessary to consider the term \( - (s_{3}^{{}} + B_{1}^{{}}g_{0}^{{}}{{\beta }_{3}})\). Taking into account the result of section 4, in this variant we obtain \(R_{1}^{{}} = B_{1}^{{}} = 0\). That is, the quadratic form \(B_{1}^{{}}\nu _{1}^{2} + B_{2}^{{}}\nu _{2}^{2} + B_{3}^{{}}\nu _{3}^{2}\) becomes degenerate \(B_{3}^{{}}\nu _{3}^{2}\). This result is of no interest for the dynamics of a gyrostat under the action of potential and gyroscopic forces.

6 CONCLUSIONS

When considering the conditions for the existence of semi-regular precessions of the second type of gyrostat under the action of potential and gyroscopic forces, a new solution of the equations of motion is constructed, in which the property of variability of the gyrostatic moment is taken into account. This solution is characterized by three invariant relations: (1.8) and formulas (1.14). The key conditions for the existence of this solution are equalities (2.8), (2.9), which characterize the mass distribution, which can be attributed to the generalized S.V. Kovalevskaya conditions. The solution is described by elementary functions of time, and the carrier body precession rate is a linear-fractional function of the trigonometric function \({\text{sin}}\,g_{0}^{{}}t\). In the general case, the solution of the problem posed is quite difficult. It can be described as follows: at the first stage of studying the existence conditions, based on the second integral from (1.3), using the invariant relation (1.8) and solving (1.14), (1.15) (in a particular case, instead of (1.14), (1.15) we can draw (1.16)) the function \(\lambda _{3}^{{}}{\text{(}}\psi {\text{)}}\) is defined; at the second stage, this function is substituted into equations (2.4)–(2.6) and three differential equations for the function \(\varepsilon {\text{(}}\psi {\text{)}}\) are found (obviously, they will be dependent); at the third stage, the problem of conditions for integrating the obtained equations in quadratures is studied.