Integrability of truncated Hugoniot-Maslov chains for trajectories of mesoscale vortices on shallow water

The problem of trajectories of "large" (mesoscale) shallow-water vortices m anifests integrability properties. The Maslow hypothesis states that such v ortices can be generated using solutions with weak pointlike singularities of the type of the square root of a quadratic form; such square-root singul ar solutions may describe the propagation of mesoscale vortices in the atmo sphere (typhoons and cyclones). Such solutions are necessarily described by infinite systems of ordinary differential equations (chains) in the Taylor coefficients of solutions in the vicinities of singularities. A proper tru ncation of the "vortex chain" for a shallow-water system is a system of 17 nonlinear equations. This sr stem becomes the Hill equation when the Coriol is force is constant and almost becomes the physical pendulum equations whe n the Coriolis force depends on the latitude. In a rough approximation, we carl then explicitly describe possible trajectories of mesoscale vortices, which are analogous to oscillations of a rotating solid body swinging on an elastic thread.