Brownian rotation in the plane ($Z_t=R(B_t)Z_0$)

The SDE $dX=-YdB-Xdt/2$, $dY=XdB-Ydt/2$ has the unique strong solution obtained by rotating $(X_0,Y_0)$ through angle $B_t$. Its coefficients are globally Lipschitz with linear growth, and $X_t^2+Y_t^2=X_0^2+Y_0^2$. The origin is absorbing and every other solution remains on its initial circle.