Commuting quadratic Hamiltonians with velocity dependent potentials

In a previous paper we studied nonhomogeneous systems of hydrodynamic type, related to commuting quadratic Hamiltonians with electromagnetic term, giv ing just a few specific examples. We now consider the problem of characteri sing the full class of systems arising in our framework, which is equivalen t to classifying all quadratic Hamiltonians with electromagnetic term, havi ng a quadratic first integral. We give a complete solution for diagonal sys tems with 2 degrees of freedom, subject to only one differential constraint .