  The indecomposable solvable Lie algebras with graded nilradical of maximal
nilindex and a Heisenberg subalgebra of codimension one are analyzed, and their
generalized Casimir invariants calculated. It is shown that rank one solvable
algebras have a contact form, which implies the existence of an associated
dynamical system. Moreover, due to the structure of the quadratic Casimir
operator of the nilradical, these algebras contain a maximal non-abelian
quasi-classical Lie algebra of dimension $2n-1$, indicating that gauge theories
(with ghosts) are possible on these subalgebras.
