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, giving just a few specific examples. We now consider the problem of characterising the full class of systems arising in our framework, which is equivalent to classifying all quadratic Hamiltonians with electromagnetic term, having a quadratic first integral. We give a complete solution for diagonal systems with 2 degrees of freedom, subject to only one differential constraint.