Calculation of Integrals of the Hugoniot–Maslov Chain for Singular Vortical Solutions of the Shallow-Water Equation

We discuss the problems of the Hugoniot–Maslov chain integrability for singular vortical solutions of the shallow-water equations on the $\beta$ plane. We show that the complex variables used to derive the chain automatically give most of the integrals of the complete and the truncated chains. We also study how some of these integrals are related to the Lagrangian invariant (potential vorticity). We discuss how to choose solutions of the chain that can be used to describe the actual trajectories of tropical cyclones.