Unitary local invariance

We address unitary local (UL) invariance of bipartite pure states. Given a bipartite state $|\Psi>>=\sum_{ij} \psi_{ij}\: |i>_1\otimes |j>_2$ the complete characterization of the class of local unitaries $U_1\otimes U_2$ for which $U_1\otimes U_2 |\Psi>>=|\Psi>>$ is obtained.The two relevant parameters are the rank of the matrix $\Psi$, $[\Psi]_{ij}=\psi_{ij}$, and the number of its equal singular values, {\em i.e.} the degeneracy of the eigenvalues of the partial traces of $|\Psi>>$.