  The unusual superconducting state in Sr$_2$RuO$_4$ has long been viewed as
being analogous to a superfluid state in liquid $^3$He. Nevertheless,
calculations based on this odd-parity state are presently unable to completely
reconcile the properties of Sr$_2$RuO$_4$. Using a self-consistent quantum
many-body scheme that employs realistic parameters, we are able to model
several signature properties of the normal and superconducting states of
Sr$_2$RuO$_4$. We find that the dominant component of the model superconducting
state is of even parity and closely related to superconducting state for the
high-$T_c$ cuprates although a smaller odd-parity component is induced by
spin-orbit coupling. This mixed pairing state gives a more complete
representation of the complex phenomena measured in Sr$_2$RuO$_4$.
