1. An arbitrary curvilinear three-web does not, generally speaking, admit automorphisms. A parallel three-web admits a three-parameter group of automorphisms. Indeed, a parallel three-web is given by the equation \(z = x + y\). Its automorphisms have the form \(x = a\tilde {x} + {{b}_{1}}\), \(y = a\tilde {y} + {{b}_{2}}\), and the corresponding autotopies \(({{A}_{1}},{{A}_{2}},{{A}_{3}})\) of a three-web (that is, admissible transformations of the parameters of the families forming the web) are \(x = a\tilde {x} + {{b}_{1}}\), \(y = a\tilde {y} + {{b}_{2}}\), \(z = a\tilde {z} + {{b}_{1}} + {{b}_{2}}\). It follows that any regular web (that is, a web locally diffeomorphic to a parallel one) also admits a three-parameter group of automorphisms. Therefore, webs that admit smaller families of automorphisms are of interest.

Three-webs with automorphisms have been studied by many authors, starting with Cartan [1]. We do review this topic and just mention two of the latest works in this direction [2, 3].

The following statements were proven in [4].

Theorem 1. If a curvilinear three-web on the real plane admits a one-parameter family of automorphisms (\(AW\)-webs), then in some local coordinates its equation can be reduced to the form

$$z = x + y + \lambda (x - y).$$
(1)

Here, \(\lambda (x - y)\) is an arbitrary smooth function of the variable \(x - y\).

Theorem 2. If a curvilinear three-web on the real plane admits a two-parameter family of automorphisms, then it is regular.

The proofs in [4] were based on the simple fact that the absolute invariants of a web are constant along the trajectories of automorphisms. In the current paper, we prove Theorem 1 in a different way, directly integrating the corresponding system of differential equations, and also show that the trajectories of automorphisms of an \(AW\)-web are geodesics of the Chern connection of a three-web. In addition, in this paper we found all \(AW\)-webs for which one of the covariant derivatives (with respect to the canonical Chern connection of this web) is equal to zero.

2. Let \(W\) be an arbitrary curvilinear three-web formed in a certain region \(D\) of the plane by lines \(x = \operatorname{const} \), \(y = \operatorname{const} \), and \(f(x,y) = \operatorname{const} \), then the equation of this web has the form \(z = f(x,y).\) We put

$${{\omega }_{1}} = {{f}_{x}}{\kern 1pt} dx,\quad {{\omega }_{2}} = {{f}_{y}}{\kern 1pt} dy.$$
(2)

By performing exterior differentiation of the forms \({{\omega }_{1}}\) and \({{\omega }_{2}}\), we get

$$d{{\omega }_{1}} = {{f}_{{xy}}}dy \wedge dx = \Gamma {{\omega }_{1}} \wedge {{\omega }_{2}},\quad d{{\omega }_{2}} = {{f}_{{xy}}}dx \wedge dy = \Gamma {{\omega }_{2}} \wedge {{\omega }_{1}},$$
(3)

where

$$\Gamma = - \frac{{{{f}_{{xy}}}}}{{{{f}_{x}}{{f}_{y}}}}.$$

Putting \(\omega = \Gamma ({{\omega }_{1}} + {{\omega }_{2}}),\) we rewrite equalities (3) as

$$d{{\omega }_{1}} = {{\omega }_{1}} \wedge \omega ,\quad d{{\omega }_{2}} = {{\omega }_{2}} \wedge \omega .$$
(4)

Exterior differentiation of the form \(\omega \) leads to the equality

$$d\omega = b{{\omega }_{1}} \wedge {{\omega }_{2}},$$
(5)

where

$$b = \frac{{{{\Gamma }_{x}}}}{{{{f}_{x}}}} - \frac{{{{\Gamma }_{y}}}}{{{{f}_{y}}}}.$$

Equations (4) and (5) are called structure equations of a three-web \(W\), and the function \(b\) is referred to as the curvature of this web. The condition \(b = 0\) characterizes the class of regular webs.

On the other hand, Eqs. (4) and (5) are equations of some torsion-free affine connection, and the forms of connection are given by

$$({{\omega }_{1}},{{\omega }_{2}}),\quad \left( {\begin{array}{*{20}{c}} \omega &0 \\ 0&\omega \end{array}} \right).$$

This connection is called the Chern connection.

The geodesic lines of the Chern connection are given by the equations

$$d{{\omega }_{1}} + {{\omega }_{1}}\omega = \Theta {{\omega }_{1}},\quad d{{\omega }_{2}} + {{\omega }_{2}}\omega = \Theta {{\omega }_{2}},$$
(6)

where \(d\) is the ordinary differentiation and \(\Theta \) is some 1-form depending on the choice of parameter on the geodesic. Excluding \(\Theta \) from Eqs. (6), we arrive at the geodesic equation in another form:

$${{\omega }_{2}}d{{\omega }_{1}} - {{\omega }_{1}}d{{\omega }_{2}} = 0.$$

From here we get \({{\omega }_{1}} = C{{\omega }_{2}},\) \(C = \operatorname{const} ,\) or

$${{f}_{x}}dx = C{{f}_{y}}dy.$$
(7)

At \(C = 0,\infty , - 1\) we get lines of the first, second, and third families of the web \(W\). At \(C = 1\) we obtain a family of geodesic lines, which, together with the lines of the third family, harmonically divide (at each point) a pair of lines of the first and second families. We can also say that the family \(C = 1\) is associated with the third family of web lines relative to the pair of the first two. Therefore, we call this family of lines 3-conjugate.

3. Automorphism of a three-web \(W\) is a local diffeomorphism of a domain \(D\), which translates the lines of the web \(W\) again into the line of this web. The proof of the following statement can be found in ([6], Theorem 6.8). Local automorphisms of a three-web \(W\) are also automorphisms of the corresponding Chern connection. Conversely, let \(\varphi \) be an automorphism of the Chern connection of some three-web \(W\), defined in the domain \(D\), and there exists a point \(p\) in \(D\) such that the differential \({{\left. {d\varphi } \right|}_{p}}\) translates tangents to lines of the web \(W\) passing through the point \(p\) into tangents to the corresponding lines passing through the point \(\varphi (p)\). Then \(\varphi \) is an automorphism of the three-web \(W\).

(Recall that the automorphism of an affine connection \(\Gamma \) defined on a manifold \(X\) is a diffeomorphism of this manifold that preserves the law of parallel translation, that is, preserves the covariant differential with respect to this connection.)

Proof of Theorem 1. Suppose that the three-web \(W\) admits a one-parameter family of automorphisms defined by a vector field \(\xi ({{\xi }_{1}},{{\xi }_{2}})\). Then the quantities \({{\xi }_{1}},{{\xi }_{2}}\) satisfy the following equations from [5]:

$$d{{\xi }_{1}} + {{\xi }_{1}}\omega = \xi {{\omega }_{1}},\quad d{{\xi }_{2}} + {{\xi }_{2}}\omega = \xi {{\omega }_{2}},$$

or, by virtue of (2) and (3),

$$d{{\xi }_{1}} + {{\xi }_{1}}\left( {\frac{{ - {{f}_{{xy}}}}}{{{{f}_{y}}}}dx - \frac{{{{f}_{{xy}}}}}{{{{f}_{x}}}}dy} \right) = \xi {{f}_{x}}dx,\quad d{{\xi }_{2}} + {{\xi }_{2}}\left( {\frac{{ - {{f}_{{xy}}}}}{{{{f}_{y}}}}dx - \frac{{{{f}_{{xy}}}}}{{{{f}_{x}}}}dy} \right) = \xi {{f}_{y}}dy.$$

Hence,

$$\begin{gathered} \frac{{\partial {{\xi }_{1}}}}{{\partial x}} = {{\xi }_{1}}\frac{{{{f}_{{xy}}}}}{{{{f}_{y}}}} + \xi {{f}_{x}},\;\;\frac{{\partial {{\xi }_{1}}}}{{\partial y}} = {{\xi }_{1}}\frac{{{{f}_{{xy}}}}}{{{{f}_{x}}}}, \\ \frac{{\partial {{\xi }_{2}}}}{{\partial x}} = {{\xi }_{2}}\frac{{{{f}_{{xy}}}}}{{{{f}_{y}}}},\;\;\frac{{\partial {{\xi }_{2}}}}{{\partial y}} = {{\xi }_{2}}\frac{{{{f}_{{xy}}}}}{{{{f}_{x}}}} + \xi {{f}_{y}}. \\ \end{gathered} $$

We put \({{\xi }_{1}} = \alpha {{f}_{x}}\), \({{\xi }_{2}} = \beta {{f}_{y}}\). Substituting into the previous equations, we get

$${{\alpha }_{y}} = 0,\;\;{{\beta }_{x}} = 0,\;\;{{\alpha }_{x}}{{f}_{x}} + \alpha {{f}_{{xx}}} = \alpha {{f}_{x}}\frac{{{{f}_{{xy}}}}}{{{{f}_{y}}}} + \xi {{f}_{x}},\;\;{{\beta }_{y}}{{f}_{y}} + \beta {{f}_{{yy}}} = \beta {{f}_{y}}\frac{{{{f}_{{xy}}}}}{{{{f}_{x}}}} + \xi {{f}_{y}}.$$

This implies \(\alpha = \alpha (x)\), \(\beta = \beta (y)\) and

$${{f}_{y}}{{(\alpha {{f}_{x}} + \beta {{f}_{y}})}_{x}} = {{f}_{x}}{{(\alpha {{f}_{x}} + \beta {{f}_{y}})}_{y}}.$$
(8)

Lemma. By variable replacements \(x = x(\tilde {x})\), \(y = y(\tilde {y})\) the functions \(\alpha (x)\) and \(\beta (y)\) can be reduced to unity.

Proof. Let us denote \(f(x(\tilde {x}),y(\tilde {y})) = \tilde {f}(\tilde {x},\tilde {y})\), then

$${{\tilde {f}}_{{\tilde {x}}}} = {{f}_{x}}\frac{{dx}}{{d\tilde {x}}},\;\;{{\tilde {f}}_{{\tilde {y}}}} = {{f}_{y}}\frac{{dy}}{{d\tilde {y}}}.$$

We put \(dx{\text{/}}d\tilde {x} = \alpha (x)\), \(dy{\text{/}}d\tilde {y} = \beta (y)\) and substitute it into (8). After transformations we arrive at the equation

$${{\tilde {f}}_{{\tilde {y}}}}{{({{\tilde {f}}_{{\tilde {x}}}} + {{\tilde {f}}_{{\tilde {y}}}})}_{{\tilde {x}}}} = {{\tilde {f}}_{{\tilde {x}}}}{{({{\tilde {f}}_{{\tilde {x}}}} + {{\tilde {f}}_{{\tilde {y}}}})}_{{\tilde {y}}}}.$$
(9)

Let us continue the proof, assuming that the indicated change of variables has been made and omit the tilde over the variables. After some calculations, we reduce Eq. (9) to the form

$${{\left( {\frac{{{{f}_{x}}}}{{{{f}_{y}}}}} \right)}_{x}} + {{\left( {\frac{{{{f}_{x}}}}{{{{f}_{y}}}}} \right)}_{y}} = 0,$$

which leads to

$$\frac{{{{f}_{x}}}}{{{{f}_{y}}}} = \varphi (x - y),$$
(10)

where \(\varphi (x - y)\) is a smooth function of the variable \(x - y\). Let us further put \(f(x,y)\) = \(g(u,{v})\), \(u = x + y\), \({v} = x - y\), then Eq. (10) becomes

$${{g}_{u}} + {{g}_{{v}}} = ({{g}_{u}} - {{g}_{{v}}})\varphi ({v}),$$

or \({{g}_{{v}}} = {{g}_{u}}\nu ({v}).\) We have

$$dg = {{g}_{u}}du + {{g}_{{v}}}d{v} = {{g}_{u}}(du + \nu ({v})d{v}) = {{g}_{u}}d(u + \lambda ({v})).$$

As a result, the web equation \(z = f(x,y)\) takes the form \(z = g(u + \lambda ({v}))\) = \(f(x + y + \lambda (x - y)\). After admissible parameter replacement \({{f}^{{ - 1}}}(z) \to z\) we arrive at Eq. (1).

\(\square \)

Because \(\alpha = \beta = 1\), it is true that \(\xi = ({{\xi }_{1}},{{\xi }_{2}}) = ({{f}_{x}},{{f}_{y}})\). Automorphisms of the \(AW\)-web have the form \(x \to x + a\), \(y \to y + a\). The trajectories of the automorphisms are the lines \(x - y = {\text{const}}\).

Theorem 3. Trajectories of automorphisms of the \(AW\)-web are geodesic and coincide with its \(3\)-conjugate family only if the three-web is regular.

Proof. From Eq. (1) we find \({{f}_{x}} = 1 + \lambda {\kern 1pt} '(x - y)\), \({{f}_{y}} = 1 - \lambda {\kern 1pt} '(x - y)\), so the geodesic equation (7) for a three-web \(AW\) becomes

$$(1 + \lambda {\kern 1pt} ')dx - C(1 - \lambda {\kern 1pt} ')dy = 0,$$

where prime means the derivative with respect to the variable \({v} = x - y\). On the trajectory of automorphisms \(x - y = \operatorname{const} \) we have \(\lambda {\kern 1pt} '(x - y) = \operatorname{const} \). We can choose the constant \(C\) so that the geodesic equation takes the form \(dx - dy = 0\) or \(x - y = {\text{const}}\). The first part of Theorem 3 is proven.

The equation of the 3-conjugate family for the \(AW\)-web transforms to

$$(1 + \lambda {\kern 1pt} ')dx - (1 - \lambda {\kern 1pt} ')dy = 0.$$

This equation coincides with the equation for the trajectories of automorphism \(dx - dy = 0\) only in case \(\lambda {\kern 1pt} ' = 0\) or \(\lambda = {\text{const}}\). But then Eq. (1) defines a regular three-web.

\(\square \)

4. Let us find the covariant derivatives of the curvature of web (1), denoting it, as above, by \(AW\). First, we calculate the curvature \(b\). We have

$${{f}_{x}} = 1 + \lambda {\kern 1pt} ',\;\;{{f}_{y}} = 1 - \lambda {\kern 1pt} ',\;\;{{f}_{{xy}}} = - \lambda {\kern 1pt} '',\;\;\Gamma = \frac{{\lambda {\kern 1pt} ''}}{{1 - {{{(\lambda {\kern 1pt} ')}}^{2}}}} = \frac{1}{2}\left( {\ln \frac{{1 + \lambda {\kern 1pt} '}}{{1 - \lambda {\kern 1pt} '}}} \right){\kern 1pt} '.$$

Further,

$${{\Gamma }_{x}} = - {{\Gamma }_{y}} = \frac{1}{2}\left( {\ln \frac{{1 + \lambda {\kern 1pt} '}}{{1 - \lambda {\kern 1pt} '}}} \right),$$
$$b = \left( {\ln \frac{{1 + \lambda {\kern 1pt} '}}{{1 - \lambda {\kern 1pt} '}}} \right){{(1 - {{(\lambda {\kern 1pt} ')}^{2}})}^{{ - 1}}}.$$
(11)

We find the covariant derivatives of the curvature \(b\) from the formula

$$db - 2b\omega = {{b}_{1}}{{\omega }_{1}} + {{b}_{2}}{{\omega }_{2}},$$

which is a differential continuation of the structure equation (5). We have

$$db = {{b}_{x}}dx + {{b}_{y}}dy = b{\kern 1pt} '{\kern 1pt} dx - b{\kern 1pt} '{\kern 1pt} dy = b{\kern 1pt} '\left( {\frac{{{{\omega }_{1}}}}{{{{f}_{x}}}} - \frac{{{{\omega }_{2}}}}{{{{f}_{y}}}}} \right),\quad 2b\omega = 2b\Gamma ({{\omega }_{1}} + {{\omega }_{2}}).$$

Consequently,

$${{b}_{1}} = \frac{{b{\kern 1pt} '}}{{{{f}_{x}}}} - 2b\Gamma = \frac{{b{\kern 1pt} '}}{{1 + \lambda {\kern 1pt} '}} - b\left( {\ln \frac{{1 + \lambda {\kern 1pt} '}}{{1 - \lambda {\kern 1pt} '}}} \right),\;\;{{b}_{2}} = - \frac{{b{\kern 1pt} '}}{{{{f}_{y}}}} - 2b\Gamma = - \frac{{b{\kern 1pt} '}}{{1 - \lambda {\kern 1pt} '}} - b\left( {\ln \frac{{1 + \lambda {\kern 1pt} '}}{{1 - \lambda {\kern 1pt} '}}} \right).$$
(12)

5. Let us find three-webs \(AW\) for which \({{b}_{1}} = 0\). According to ([7], p. 69) this condition distinguishes three-webs \({{B}_{1}}\) formed by a family of parallel lines and integral curves of two Riccati equations of a special form. Let us show that such a class exists, and the solution can be found in quadratures. Note that, to describe this class, we cannot use the results from [7], because the moving frame there was normalized by the condition that one of the covariant derivatives of the curvature is equal to unity, while in this article the normalization is different—the trajectories of automorphisms are written in the form \(x - y = {\text{const}}\).

As can be seen from (12), the condition \({{b}_{1}} = 0\) leads to the equation

$$\frac{{b{\kern 1pt} '}}{{1 + \lambda {\kern 1pt} '}} - b\left( {\ln \frac{{1 + \lambda {\kern 1pt} '}}{{1 - \lambda {\kern 1pt} '}}} \right) = 0 \Rightarrow \frac{{b{\kern 1pt} '}}{b} = \frac{{2\lambda {\kern 1pt} ''}}{{1 - \lambda {\kern 1pt} '}},$$

or \((\ln b){\kern 1pt} ' = (\ln {{(1 - \lambda {\kern 1pt} ')}^{{ - 2}}}){\kern 1pt} '\). Hence,

$$b = \frac{\alpha }{{2{{{(1 - \lambda {\kern 1pt} ')}}^{2}}}},\;\;\alpha = \operatorname{const} .$$

If \(\alpha = 0\), then \(b = 0\), and we get a regular web. Let further \(\alpha \ne 0\). Comparing the last equality with (11), we arrive at the relation

$$\frac{\alpha }{2}\frac{{1 + \lambda {\kern 1pt} '}}{{1 - \lambda {\kern 1pt} '}} = \left( {\ln \frac{{1 + \lambda {\kern 1pt} '}}{{1 - \lambda {\kern 1pt} '}}} \right),$$

or

$$t{\kern 1pt} '' = \frac{1}{2}\alpha {{e}^{t}},$$
(13)

where

$$t = \ln \frac{{1 + \lambda {\kern 1pt} '}}{{1 - \lambda {\kern 1pt} '}}.$$
(14)

In (13) we put

$$t{\kern 1pt} '({v}) = \Theta (t({v})) \Rightarrow t{\kern 1pt} ''(v) = \Theta {\kern 1pt} '(t)t{\kern 1pt} '({v})) = \Theta {\kern 1pt} '(t)\Theta .$$

As a result, (13) takes the form \(2\Theta d\Theta = \alpha {{e}^{t}}dt.\) Therefore, \({{\Theta }^{2}} = \alpha {{e}^{t}} + \beta \), \(\beta = \operatorname{const} ,\) and

$$\Theta = \frac{{dt}}{{dv}} = \sqrt {\alpha {{e}^{t}} + \beta } \Rightarrow {v} = \int \frac{{dt}}{{\sqrt {\alpha {{e}^{t}} + \beta } }}.$$

Replacement \(\alpha {{e}^{t}} + \beta = {{u}^{2}}\) reduces the integral to the form

$${v} = 2\int \frac{{du}}{{{{u}^{2}} - \beta }}.$$
(15)

Case 1. \(\beta = {{\delta }^{2}} \ne 0\), then

$${v} = \frac{1}{\delta }\ln \left| {\frac{{\sqrt {\alpha {{e}^{t}} + {{\delta }^{2}}} - \delta }}{{\sqrt {\alpha {{e}^{t}} + {{\delta }^{2}}} + \delta }}} \right| - \frac{\gamma }{\delta },\;\;\gamma = \operatorname{const} .$$

After simple transformations, \(\alpha > 0\) get for

$${{e}^{t}} = \frac{{4{{\delta }^{2}}}}{\alpha }\frac{{{{e}^{{\delta {v} + \gamma }}}}}{{{{{({{e}^{{\delta {v} + \gamma }}} - 1)}}^{2}}}},$$

for \(\alpha < 0\)

$${{e}^{t}} = - \frac{{4{{\delta }^{2}}}}{\alpha }\frac{{{{e}^{{\delta {v} + \gamma }}}}}{{{{{({{e}^{{\delta {v} + \gamma }}} - 1)}}^{2}}}}.$$

As we can see, these cases are identical, so we consider only the first option below. It follows from (14) that

$$\lambda {\kern 1pt} ' = \frac{{{{e}^{t}} - 1}}{{{{e}^{t}} + 1}}.$$

Substituting here the previous expression for \({{e}^{t}}\), we find

$$\lambda {\kern 1pt} ' = \frac{{\varepsilon - {{{\sinh }}^{2}}\left( {\frac{{\delta {v} + \gamma }}{2}} \right)}}{{\varepsilon + {{{\sinh }}^{2}}\left( {\frac{{\delta {v} + \gamma }}{2}} \right)}},\;\;\varepsilon = \frac{{{{\delta }^{2}}}}{\alpha }.$$
(16)

Next, we put \(\frac{{\delta {v} + \gamma }}{2} = X\) and in the right-hand side of (16) we express \({{\sinh }^{2}}X\) through \(\tanh X\). As a result, we get

$$\lambda {\kern 1pt} ' = \frac{{\varepsilon - (\varepsilon + 1){{{\tanh }}^{2}}X}}{{\varepsilon - (\varepsilon - 1){{{\tanh }}^{2}}X}},$$
$$\lambda = \int \frac{{\varepsilon - (\varepsilon + 1){{{\tanh }}^{2}}X}}{{\varepsilon - (\varepsilon - 1){{{\tanh }}^{2}}X}}d{v} = \frac{2}{\delta }\int \frac{{\varepsilon - (\varepsilon + 1){{{\tanh }}^{2}}X}}{{\varepsilon - (\varepsilon - 1){{{\tanh }}^{2}}X}}dX.$$

After standard replacement \(\tanh X = z\) we arrive at the integral

$$\lambda = \frac{2}{\delta }\int \frac{{\varepsilon - (\varepsilon + 1){{z}^{2}}}}{{\varepsilon - (\varepsilon - 1){{z}^{2}}}}\frac{{dz}}{{1 - {{z}^{2}}}} = \frac{{4\varepsilon }}{{\delta (\varepsilon - 1)}}\int \frac{1}{{\frac{\varepsilon }{{\varepsilon - 1}} - {{z}^{2}}}}dz - \frac{2}{\delta }\int \frac{{dz}}{{1 - {{z}^{2}}}}.$$
(17)

Subcase 1a. \(\frac{\varepsilon }{{\varepsilon - 1}} = {{\eta }^{2}}\), and \(\eta \ne 0\), because \(\varepsilon = {{\delta }^{2}}{\text{/}}\alpha \ne 0\). Then we have

$$\begin{aligned} \lambda = \frac{{4{{\eta }^{2}}}}{\delta }\int \frac{{dz}}{{{{\eta }^{2}} - {{z}^{2}}}} - \frac{2}{\delta }\int dX = - \frac{{2\eta }}{\delta }\ln \left| {\frac{{\tanh X - \eta }}{{\tanh X + \eta }}} \right| - \frac{2}{\delta }X + {{C}_{1}} \\ = - \frac{{2\eta }}{\delta }\ln \left| {\frac{{\tanh \frac{{\delta {v} + \gamma }}{2} - \eta }}{{\tanh \frac{{\delta {v} + \gamma }}{2} + \eta }}} \right| - {v} + C. \\ \end{aligned} $$
(18)

Subcase 1b. At \(\frac{\varepsilon }{{\varepsilon - 1}} = - {{\eta }^{2}}\) from (17) we obtain

$$\begin{aligned} \lambda = - \frac{{4{{\eta }^{2}}}}{\delta }\int \frac{{dz}}{{ - {{\eta }^{2}} - {{z}^{2}}}} - \frac{2}{\delta }\int dX = \frac{{4\eta }}{\delta }\arctan \frac{{\tanh X}}{\eta } - \frac{2}{\delta }X + {{C}_{1}} \\ = \frac{{4\eta }}{\delta }\arctan \frac{{\tanh \frac{{\delta {v} + \gamma }}{2}}}{\eta } - {v} + C. \\ \end{aligned} $$
(19)

Case 2. \(\beta = - {{\delta }^{2}} \ne 0\). Then it follows from (15) that

$${v} = \frac{2}{\delta }\arctan \frac{u}{\delta } - \frac{\gamma }{\delta },$$

which leads to

$${{e}^{t}} = \frac{{{{\delta }^{2}}}}{\alpha }{{\cos }^{{ - 2}}}\left( {\frac{{\delta {v} + \gamma }}{2}} \right).$$

Putting \(\frac{{\delta {v} + \gamma }}{2} = X\) and \(\frac{{{{\delta }^{2}}}}{\alpha } = \eta \), we have

$${{e}^{t}} = \eta (1 + \mathop {\tan }\nolimits^2 X),\;\;\lambda {\kern 1pt} ' = \frac{{{{{\tan }}^{2}}X + \frac{{\eta - 1}}{\eta }}}{{{{{\tan }}^{2}}X + \frac{{\eta + 1}}{\eta }}}.$$

Subcase 2a. \((\eta + 1){\text{/}}\eta = - {{\kappa }^{2}} \ne 0\) \( \Rightarrow \)

$$\lambda {\kern 1pt} ' = \frac{{{{{\tan }}^{2}}X + 2 + {{\kappa }^{2}}}}{{{{{\tan }}^{2}}X - {{\kappa }^{2}}}}.$$

We put \(\tan X = z\), then

$$\lambda = \int \frac{{{{z}^{2}} + 2 + {{\kappa }^{2}}}}{{{{z}^{2}} - {{\kappa }^{2}}}}d{v} = \frac{2}{\delta }\int \frac{{{{z}^{2}} + 2 + {{\kappa }^{2}}}}{{{{z}^{2}} - {{\kappa }^{2}}}}dX = \frac{2}{\delta }\int \frac{{{{z}^{2}} + 2 + {{\kappa }^{2}}}}{{{{z}^{2}} - {{\kappa }^{2}}}}\frac{{dz}}{{{{z}^{2}} + 1}}.$$

However,

$$\frac{{{{z}^{2}} + 2 + {{\kappa }^{2}}}}{{({{z}^{2}} - {{\kappa }^{2}})({{z}^{2}} + 1)}} = \frac{2}{{{{z}^{2}} - {{\kappa }^{2}}}} - \frac{1}{{{{z}^{2}} + 1}},$$

Hence,

$$\begin{aligned} \lambda = \frac{2}{\delta }\int \left( {\frac{2}{{{{z}^{2}} - {{\kappa }^{2}}}} - \frac{1}{{{{z}^{2}} + 1}}} \right)dz\frac{2}{\delta }\left( {\frac{1}{\kappa }\ln \left| {\frac{{z - \kappa }}{{z + \kappa }}} \right| - X} \right) + C \\ = \frac{2}{{\delta \kappa }}\ln \left| {\frac{{\tan \frac{{\delta {v} + \gamma }}{2} - \kappa }}{{\tan \frac{{\delta {v} + \gamma }}{2} + \kappa }}} \right| - \frac{2}{\delta }\frac{{\delta {v} + \gamma }}{2} + {{C}_{1}} = \frac{2}{{\delta \kappa }}\ln \left| {\frac{{\tan \frac{{\delta {v} + \gamma }}{2} - \kappa }}{{\tan \frac{{\delta {v} + \gamma }}{2} + \kappa }}} \right| - {v} + C. \\ \end{aligned} $$
(20)

Subcase 2b. \((\eta + 1){\text{/}}\eta = {{\kappa }^{2}} \ne 0\) \( \Rightarrow \)

$$\lambda {\kern 1pt} ' = \frac{{{{{\tan }}^{2}}X + 2 - {{\kappa }^{2}}}}{{{{{\tan }}^{2}}X + {{\kappa }^{2}}}} = \frac{{{{z}^{2}} + 2 - {{\kappa }^{2}}}}{{{{z}^{2}} + {{\kappa }^{2}}}},$$

which results in

$$\lambda = \int \frac{{{{z}^{2}} + 2 - {{\kappa }^{2}}}}{{{{z}^{2}} + {{\kappa }^{2}}}}d{v} = \frac{2}{\delta }\int \frac{{{{z}^{2}} + 2 - {{\kappa }^{2}}}}{{{{z}^{2}} + {{\kappa }^{2}}}}dX = \frac{2}{\delta }\int \frac{{{{z}^{2}} + 2 - {{\kappa }^{2}}}}{{{{z}^{2}} + {{\kappa }^{2}}}}\frac{{dz}}{{{{z}^{2}} + 1}}.$$

However,

$$\frac{{{{z}^{2}} + 2 - {{\kappa }^{2}}}}{{({{z}^{2}} + {{\kappa }^{2}})({{z}^{2}} + 1)}} = \frac{2}{{{{z}^{2}} + {{\kappa }^{2}}}} - \frac{1}{{{{z}^{2}} + 1}},$$

Therefore,

$$\begin{aligned} \lambda = \frac{2}{\delta }\int \left( {\frac{2}{{{{z}^{2}} + {{\kappa }^{2}}}} - \frac{1}{{{{z}^{2}} + 1}}} \right)dz = \frac{4}{{\delta \kappa }}\arctan \frac{z}{\kappa } - \frac{2}{\delta }X + {{C}_{1}} \\ = \frac{4}{{\delta \kappa }}\arctan \frac{{\tan \frac{{\delta v + \gamma }}{2}}}{\kappa } - {v} + C. \\ \end{aligned} $$
(21)

Subcase 2c. \(\kappa = 0\). Then

$$\lambda {\kern 1pt} ' = \frac{{{{{\tan }}^{2}}X + 2}}{{{{{\tan }}^{2}}X}}$$

and

$$\lambda = \frac{2}{\delta }( - X - 2\cot X) + {{C}_{1}} = - {v} - \frac{4}{\delta }\cot \frac{{\delta {v} + \gamma }}{2} + C.$$
(22)

Case 3. \(\beta = 0\). From (15) we find

$${v} = - \frac{2}{u} + \gamma = \mp \frac{2}{{\sqrt \alpha {{e}^{t}}}} + \gamma ,\quad \gamma = \operatorname{const} .$$

This leads to

$${{e}^{t}} = \frac{4}{{\alpha {{{({v} - \gamma )}}^{2}}}},\;\;\lambda {\kern 1pt} ' = \frac{{{{e}^{t}} - 1}}{{{{e}^{t}} + 1}} = \frac{{4 - \alpha {{{({v} - \gamma )}}^{2}}}}{{4 + \alpha {{{({v} - \gamma )}}^{2}}}}.$$

Integration yields

for \(\alpha > 0\)

$$\lambda = - {v} + \frac{4}{{\sqrt \alpha }}\arctan \left( {\frac{{\sqrt \alpha }}{2}({v} - \gamma )} \right) + C;$$
(23)

for \(\alpha < 0\)

$$\lambda = - {v} - \frac{2}{{\sqrt { - \alpha } }}\ln \left| {\frac{{\sqrt { - \alpha } ({v} - \gamma ) - 2}}{{\sqrt { - \alpha } ({v} - \gamma ) + 2}}} \right| + C.$$
(24)

We have proved the following theorem.

Theorem 4. There are seven types of curvilinear three-webs admitting a one-parameter family of infinitesimal automorphisms for which one of the covariant derivatives of the curvature is equal to zero. These are three-webs defined by equations of the form (1), where the function \(\lambda \) is calculated by formulas (18)–(24).