О точных решениях системы квазилинейных уравнений, описывающей интегрируемые геодезические потоки на поверхности

In this paper, for the first time, explicit solutions of a semi-Hamiltonian system of quasi-linear differential equations by the generalized hodograph method are found. These solutions define (local) metrics on a surface for which the geodesic flow has a polynomial in momenta integrals of the fourth degree.