We provide a general solution for a first order ordinary differential equation with a rational right-hand side, which arises in constructing asymptotics for large time of simultaneous solutions of the Korteweg-de Vries equation and the stationary par
t of its higher non-autonomous symmetry. This symmetry is determined by a linear combination of the first higher autonomous symmetry of the Korteweg-de Vries equation and of its classical Galileo symmetry. This general solution depends on an arbitrary parameter. By the implicit function theorem, locally it is determined by the first integral explicitly written in terms of hypergeometric functions. A particular case of the general solution defines self-similar solutions of the Whitham equations, found earlier by G.V. Potemin in 1988. In the well-known works by A.V. Gurevich and L.P. Pitaevsky in early 1970s, it was established that these solutions of the Whitham equations describe the origination in the leading term of non-damping oscillating waves in a wide range of problems with a small dispersion. The result of this article supports once again an empirical rule saying that under various passages to the limits, integrable equations can produce only integrable, in certain sense, equations. We propose a general conjecture: integrable ordinary differential equations similar to that considered in the present paper should also arise in describing the asymptotics at large times for other symmetry solutions to evolution equations admitting the application of the method of inverse scattering problem.
The effect of the small dispersion on the self-focusing of solutions of the equations of nonlinear geometric optics in one-dimensional case is investigated. In the main order this influence is described by means of the universal special solution of t
he nonlinear Schrodinger equation, which is isomonodromic. Analytic and asymptotic properties of this solution are described.
We construct solutions of analogues of the nonstationary Schrodinger equation corresponding to the polynomial isomonodromic Hamiltonian Garnier system with two degrees of freedom. This solutions are obtained from solutions of systems of linear ordina
ry differential equations whose compatibility condition is the Garnier system. This solutions upto explicit transform also satisfy the Belavin --- Polyakov --- Zamolodchikov equations with four time variables and two space variables.
