  We develop a general technique for finding self-adjoint extensions of a
symmetric operator that respect a given set of its symmetries. Problems of this
type naturally arise when considering two- and three-dimensional Schr\"odinger
operators with singular potentials. The approach is based on constructing a
unitary transformation diagonalizing the symmetries and reducing the initial
operator to the direct integral of a suitable family of partial operators. We
prove that symmetry preserving self-adjoint extensions of the initial operator
are in a one-to-one correspondence with measurable families of self-adjoint
extensions of partial operators obtained by reduction. The general scheme is
applied to the three-dimensional Aharonov-Bohm Hamiltonian describing the
electron in the magnetic field of an infinitely thin solenoid. We construct all
self-adjoint extensions of this Hamiltonian, invariant under translations along
the solenoid and rotations around it, and explicitly find their eigenfunction
expansions.
