Breaking solitons. VI. Extension of systems of hydrodynamic type

Systems of differential equations, admitting the Lax representation and extending the systems of hydrodynamic type, connected with the Volterra model and Toda lattice, are presented. A construction of differential operator equations with derivatives of arbitrary order with respect to the variables $t$ and $y$ and possessing a reduction preserving the eigenvalues of the corresponding operator $L$ is suggested. Dynamical systems having a Lax representation and generalizing the Toda lattice are constructed. A construction of integrable Euler equations admitting a Lax representation with $n$ independent spectral parameters and connected with $n$ Riemann surfaces is found.