  Nonholonomic mechanical systems have been attracting more interest in recent
years because of their rich geometric properties and their applications in
Engineering. In all generality, we discuss the reduction of a Hamilton-Jacobi
theory for systems subject to nonholonomic constraints and that are invariant
under the action of a group of symmetries. We consider nonholonomic systems
subject to linear or nonlinear constraints, with different positioning with
respect to the symmetries. We describe the reduction procedure first, to later
reconstruct solutions in the unreduced picture, by starting from a reduced
Hamilton-Jacobi equation. Examples can be depicted in a wide range of
scenarios: from free particles with linear constraints, to vehicle motion.
