  We determine fundamental systems of invariants for complex solvable rigid Lie
algebras having nonsplit nilradicals of characteristic sequence $(3,1,..,1)$,
these algebras being the natural followers of solvable algebras having
Heisenberg nilradicals. A special case of this allows us to obtain a criterion
to determine the number of functionally independent invariants of rank one
subalgebras of (real or complex) solvable Lie algebras. Finally, we give
examples of the inverse procedure, obtaining fundamental systems of an algebra
starting from rank one subalgebras.
