  We have studied the superconducting state of Sr$_2$RuO$_4$ under a magnetic
field parallel to the superconducting plane. We show that due to a weak
spin-orbit coupling, non-unitary
k$_y(\hat{\mathbf{z}}-i\alpha\hat{\mathbf{y}})$ state is formed right at
H$_{c2}$, then changes to unitary k$_y\hat{\mathbf{z}}$ state, as a magnetic
field is lowered. In terms of this crossover, we address the origin of the
observed double peaks of specific heat and the disappearance of the double
peaks at low fields.
