Hydrodynamic-type systems and homogeneous Hamiltonian operators: a necessary condition of compatibility

Using the theory of coverings, it is presented a necessary condition to write a hydrodynamic-type system in Hamiltonian formulation. Explicit conditions for first, second and third order homogeneous Hamiltonian operators are shown. In particular, an alternative proof of Tsarev's theorem about compatibility conditions for first order operators is obtained by using this method.
Then, analogous conditions are presented for non local homogeneous Hamiltonian operators of first and third order.
Finally, it is discussed the projective invariance for second and third order operators.
The talk is based on joint work with Raffaele Vitolo arXiv:2007.15294.