  Two infinite sets of Galilean invariant equations are derived using the
irreducible representations of the orthochrous extended Galilean group. It is
shown that one set contains the Schr\"odinger equation, which is the
fundamental equation for ordinary matter, and the other set has a new
asymmetric equation, which is proposed to be the fundamental equation for dark
matter. Using this new equation, a theory of dark matter is developed and its
profound physical implications are discussed. This theory explains the
currently known properties of dark matter and also predicts a detectable
gravitational radiation.
