  Nonperturbative exact solutions are allowed for quantum integrable models in
one space-dimension. Going beyond this class we propose an alternative Lax
matrix approach, exploiting the hidden multi-time concept in integrable systems
and construct a novel quantum nonlinear Schroedinger model in quasi-two
dimensions. An intriguing field commutator is discovered, confirming the
integrability of the model and yielding its exact Bethe ansatz solution with
rich scattering and bound-state properties. The universality of the scheme is
expected to cover diverse models, opening up a new direction in the field.
