  For the one-dimensional nonlinear Schroedinger equation with a complex
potential, it is shown that if this potential is not parity-time (PT)
symmetric, then no continuous families of solitons can bifurcate out from
linear guided modes, even if the linear spectrum of this potential is all real.
Both localized and periodic non-PT-symmetric potentials are considered, and the
analytical conclusion is corroborated by explicit examples. Based on this
result, it is argued that PT-symmetry of a one-dimensional complex potential is
a necessary condition for the existence of soliton families.
