  It is proved a theorem providing necessary and sufficient conditions enabling
one to map a nonlinear system of first order partial differential equations,
polynomial in the derivatives, to an equivalent autonomous first order system
polynomially homogeneous in the derivatives. The result is intimately related
to the symmetry properties of the source system, and the proof, involving the
use of the canonical variables associated to the admitted Lie point symmetries,
is constructive. First order Monge-Amp\`ere systems, either with constant
coefficients or with coefficients depending on the field variables, where the
theorem can be successfully applied, are considered.
