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The irreducible representations of the extended Galilean group are used toderive infinite sets of symmetric and asymmetric second-order differentialequations with constant coeffcients. All derived equations are local and theirLagrangians exist. It is shown that the asymmetric equations are Galileaninvariant but the symmetric ones are not. By specifying quantum and classicalphysical settings, the constants in the equations are determined and thefundamental wave equations, including the Schr\"odinger, Schr\"odinger-like andnew asymmetric equations, are obtained; the derived wave equation isnon-fundamental. Formulation of wave theories based on the fundamental andnon-fundamental wave equaions is considered, and physical implications of thesetheories on the wave description are discussed.

Nonrelativistic Fundamental Quantum and Classical Wave Equations