One class of Liouville-type systems

We prove that a class of systems with the Lagrangian of the form L = g_{ij}(u) u_x i u_t j /2 + f(u) is of the Liouville type. We construct new integrable Hamiltonian systems related to the symmetries of the hyperbolic systems under consideration by substitutions of the Miura transformation type. For one of the systems obtained, we construct the second-order recursion operator.