  We apply various conventional tests of integrability to the supersymmetric
nonlinear Schr\"odinger equation. We find that a matrix Lax pair exists and
that the system has the Painlev\'e property only for a particular choice of the
free parameters of the theory. We also show that the second Hamiltonian
structure generalizes to superspace only for these values of the parameters. We
are unable to construct a zero curvature formulation of the equations based on
OSp(2$|$1). However, this attempt yields a nonsupersymmetric fermionic
generalization of the nonlinear Schr\"odinger equation which appears to possess
the Painlev\'e property.
