The governing PDEs for nonlinear materials science and conformal geometry are highly nonlinear PDE’s called Beltrami systems. There is little known about existence of solutions to these PDEs other than the classical results of Weyl-Schouten. Here we discuss uniqueness and analytic continuation for systems with measurable coefficients. Ultimately this boils down to the Hilbert-Smith conjecture for transformation groups (really Hilbert’s 5th problem - locally compact group acting effectively on a nice space is a Lie group). We solve this conjecture for groups which are conformal with respect to a measurable background structure (in fact we prove a stronger result). This appears to be the strongest known solution to this problem.