General dynamical systems described by first order quasilinear PDEs reducible to homogeneous and autonomous form

Within the framework of Lie group analysis of differential equations, a theorem is determined stating necessary and sufficient conditions allowing one to recover an invertible point transformation mapping a general dynamical system described by nonhomogeneous and nonautonomous first order quasilinear partial differential equations to homogeneous and autonomous form. The proof of the theorem is constructive and the new independent and dependent variables are obtained by determining the canonical variables associated to a suitable subalgebra of the Lie algebra of point symmetries admitted by the source system. The theorem is applied by considering some examples of physical interest arising from different contexts.