NONHOMOGENEOUS SYSTEMS OF HYDRODYNAMIC TYPE, RELATED TO QUADRATIC HAMILTONIANS WITH ELECTROMAGNETIC TERM

We consider a class of non-homogeneous systems of hydrodynamic type: q (t)(i) = v(j)(i)(q)q(x)(j) + phi(i)(q), i = 1, ..., n, which can be re lated to quadratic Hamiltonians with electromagnetic terms. Whilst it is unlikely that our systems are generally integrable, they do possess intriguing properties, such as (always) having a higher conservation law and (sometimes) a 2n-parameter family of exact solutions. In fact our systems coincide with those possessing a conservation law with the density L being a quadratic expression in the first derivatives: L = 1/2 (i,j = 1)Sigma(n) g(ij)q(x)(i)q(x)(j) + (k = 1)Sigma(n) A(k)q(x)(k ) - h. We discuss several examples, some of which have important appli cations.