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According to Maslov, many 2D quasilinear systems of PDE possess only three algebras of singular solutions with properties of “structural” self-similarity and stability. They are the algebras of shock waves, “narrow” solitons, and “square-root” point singularities (solitary vortices). Their propagation is described by infinite chains of ODE (the Hugoniót–Maslov chains). We consider the Hugoniót-Maslov chain for the “square-root” point singularities of the shallow water equations. We discuss different related mathematical questions (in particular, unexpected integrability effects) as well as their possible application to the problem of typhoon dynamics.