  We consider nilpotent Lie groups for which the derived subgroup is abelian.
We equip them with subRiemannian metrics and we study the normal Hamiltonian
flow on the cotangent bundle. We show a correspondence between normal
trajectories and polynomial Hamiltonians in some euclidean space. We use the
aforementioned correspondence to give a criterion for the integrability of the
normal Hamiltonian flow. As an immediate consequence, we show that in
Engel-type groups the flow of the normal Hamiltonian is integrable. For Carnot
groups that are semidirect products of two abelian groups, we give a set of
conditions that normal trajectories must fulfill to be globally
length-minimizing. Our results are based on a symplectic reduction procedure.
