  The two-dimensional Jordan-Wigner transformation is used to investigate the
zero temperature spin-Peierls transition for an anisotropic two-dimensional XY
model in adiabatic limit. The phase diagram between the dimerized (D) state and
uniform (U) state is shown in the parameter space of dimensionless interchain
coupling $h$ $(=J_{\perp}/J)$ and spin-lattice coupling $\eta$. It is found
that the spin-lattice coupling $\eta$ must exceed some critical value $\eta_c$
in order to reach the D phase for any finite $h$. The dependence of $\eta_c$ on
$h$ is given by $-1/\ln h$ for $h\to 0$ and the transition between U and D
phase is of first-order for at least $h>10^{-3}$.
