Sub-semi-Riemannian geometry on $H$-type groups

We consider $H$(eisenberg)-type groups whose law of left translation gives rise to a bracket generating distribution of step 2. In the contrast with sub-Riemannian studies we furnish the horizontal distribution with a nondegenerate indefinite metric of arbitrary index and investigate the problem concerning causal geodesics on underlying manifolds. The exact formulae for geodesics are obtained.