  We perform the complete group classification in the class of cubic
Schr\"odinger equations of the form
$i\psi_t+\psi_{xx}+\psi^2\psi^*+V(t,x)\psi=0$ where $V$ is an arbitrary
complex-valued potential depending on $t$ and $x$. We construct all possible
inequivalent potentials for which these equations have non-trivial Lie
symmetries using algebraic and compatibility methods simultaneously. Our
classification essentially amends earlier works on the subject.
