  We classify (1+3)-dimensional Pauli equations for a spin-1/2 particle
interacting with the electro-magnetic field, that are solvable by the method of
separation of variables. As a result, we obtain the eleven classes of
vector-potentials of the electro-magnetic field A(t,x) providing separability
of the corresponding Pauli equations. It is established, in particular, that
the necessary condition for the Pauli equation to be separable into
second-order matrix ordinary differential equations is its equivalence to the
system of two uncoupled Schroedinger equations. In addition, the magnetic field
has to be independent of spatial variables. We prove that coordinate systems
and the vector-potentials of the electro-magnetic field providing the
separability of the corresponding Pauli equations coincide with those for the
Schroedinger equations. Furthermore, an efficient algorithm for constructing
all coordinate systems providing the separability of Pauli equation with a
fixed vector-potential of the electro-magnetic field is developed. Finally, we
describe all vector-potentials A(t,x) that (a) provide the separability of
Pauli equation, (b) satisfy vacuum Maxwell equations without currents, and (c)
describe non-zero magnetic field.
