  We suggest an effective approach to separation of variables in the
Schr\"odinger equation with two space variables. Using it we classify
inequivalent potentials $V(x_1,x_2)$ such that the corresponding Schr\" odinger
equations admit separation of variables. Besides that, we carry out separation
of variables in the Schr\" odinger equation with the anisotropic harmonic
oscillator potential $V=k_1x_1^2+k_2x_2^2$ and obtain a complete list of
coordinate systems providing its separability. Most of these coordinate systems
depend essentially on the form of the potential and do not provide separation
of variables in the free Schr\" odinger equation ($V=0$).
