We consider the action of a Lie group that leaves invariant the sub-Riemannian structures associated to Goursat systems and Euclidean metrics. The low dimensional cases contain the well-known Heisenberg group, as well as the nilpotent groups associated to the Martinet and Engel systems. For arbitrary dimensions, we establish the complete integrability of the associated adjoint system by means of the explicit calculation of Casimir functions of the associated nilpotent Poisson algebra. We define a new algebraic curve in total space and compute two examples of sub-Riemannian extremals in cotangent space. Our approach fits certain non-holonomic classical problems, particularly that of the physical model for a plasma considered as a gas of non interacting charged particles under the influence of a static non homogeneous magnetic field.