  The stationary Gross-Pitaevskii equation in one dimension is considered with
a complex periodic potential satisfying the conditions of the PT (parity-time
reversal) symmetry. Under rather general assumptions on the potentials we prove
bifurcations of PT-symmetric nonlinear bound states from the end points of a
real interval in the spectrum of the non-selfadjoint linear Schrodinger
operator with a complex PT-symmetric periodic potential. The nonlinear bound
states are approximated by the effective amplitude equation, which bears the
form of the cubic nonlinear Schrodinger equation. In addition we provide
sufficient conditions for the appearance of complex spectral bands when the
complex $\PT$-symmetric potential has an asymptotically small imaginary part.
