Inspired by Gromov’s partial differential relations, we introduce a notion of differential transmutation, which allows to transfer some local properties of solutions of a PDE to solutions of another PDE, in particular hypoellipticity, weak and strong unique continuation properties and the Runge property. The latest refers to the possibility to approximate some given local solutions by a global solution, with preassigned singularities in the holes of the initial domain. As examples we prove that 2D Lamé-Navier
system and the 3D steady Stokes system, can be obtained as differential
transmutations of appropriate tensorizations of the Laplace operator.