  We introduce a model with conserved dynamics, where nearest neighbor pairs of
spins $\uparrow \downarrow (\downarrow \uparrow)$ can exchange to assume the
configuration $\downarrow \uparrow (\uparrow \downarrow)$, with rate $\beta
(\alpha)$, through energy decreasing moves only. We report exact solution for
the case when one of the rates, $\alpha$ or $\beta$, is zero. The
irreversibility of such dynamics results in strong dependence on the initial
conditions. Domain wall arguments suggest that for more general models with
steady states the dynamical critical exponent for the anisotropic spin exchange
is different from the isotropic value.
