  We characterize locally Lipschitz mappings and existence of Lipschitz
extensions through a first order nonlinear system of PDEs. We extend this study
to graded group-valued Lipschitz mappings defined on compact Riemannian
manifolds. Through a simple application, we emphasize the connection between
these PDEs and the Rumin complex. We introduce a class of 2-step groups,
satisfying some abstract geometric conditions and we show that Lipschitz
mappings taking values in these groups and defined on subsets of the plane
admit Lipschitz extensions. We present several examples of these groups, called
Allcock groups, observing that their horizontal distribution may have any
codimesion. Finally, we show how these Lipschitz extensions theorems lead us to
quadratic isoperimetric inequalities in all Allcock groups.
