  We introduce a class of nonlinear partial differential equations in a product
space which are at the interface of Finsler and sub-Riemannian geometry. To
such equations we associate a non-isotropic Minkowski gauge $\Theta$ for which
we introduce a suitable notion of Legendre transform $\Theta^0$. We compute the
action of the relevant nonlinear PDEs on ``radial" functions, i.e., functions
of $\Theta^0$, and by exploiting it we are able to compute explicit fundamental
solutions of such PDEs.
