  The integrability of a system of two symmetrically coupled higher-order
nonlinear Schr\"{o}dinger equations with parameter coefficients is tested by
means of the singularity analysis. It is proven that the system passes the
Painlev\'{e} test for integrability only in ten distinct cases, of which two
are new. For one of the new cases, a Lax pair and a multi-field generalization
are obtained; for the other one, the equations of the system are uncoupled by a
nonlinear transformation.
