Unitary local invariance

We address unitary local 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 in terms of the singular values of the matrix $\Psi=[\psi]_{ij}$