  We describe co-adjoint orbits and Casimir functions for two-step
free-nilpotent Lie algebras. The symplectic foliation consists of affine
subspaces of the Lie coalgebra of different dimensions.
  Further, we consider left-invariant time-optimal problems on two-step Carnot
groups, for which the set of admissible velocities is a strictly convex
compactum in the first layer of the Lie algebra containing the origin in its
interior. We describe integrals for the vertical subsystem of the Hamiltonian
system of Pontryagin maximum principle. Further, we describe constancy and
periodicity of solutions to this subsystem and controls, and characterize its
flow, for two-dimensional co-adjoint orbits.
