  The irreducible representations of the extended Galilean group are used to
derive infinite sets of symmetric and asymmetric second-order differential
equations with constant coeffcients. All derived equations are local and their
Lagrangians exist. It is shown that the asymmetric equations are Galilean
invariant but the symmetric ones are not. By specifying quantum and classical
physical settings, the constants in the equations are determined and the
fundamental wave equations, including the Schr\"odinger, Schr\"odinger-like and
new asymmetric equations, are obtained; the derived wave equation is
non-fundamental. Formulation of wave theories based on the fundamental and
non-fundamental wave equaions is considered, and physical implications of these
theories on the wave description are discussed.
