  Based on an original classification of differential equations by types of
regular Lie group actions, we offer a systematic procedure for describing
partial differential equations with prescribed symmetry groups. Using a new
powerful algebraic technique based on the so-called covariant form of a
differential equation, we give an effective algorithm for constructing
differential equations whose symmetry groups regularly and freely act on the
space of dependent and independent variables. As an application, we derive a
complete classification of quasi-linear scalar second-order partial
differential equations with regular free symmetry groups of dimension no
greater than three.
