Symmetry and Integrability of Equations of Mathematical Physics − 2011 Yuri Karadzhov (Institute of Mathematics of NAS of Ukraine, Kyiv, Ukraine) Matrix Superpotentials Abstract: The classification of matrix-valued shape-invariant super potentials which give rise to new exactly solvable systems of Schrödinger equations is presented. The superpotentials of the generic form $W_k = kQ + P +\frac1k R$, where $k$ is variable parameter, $P, Q$ and $R$ are hermitian matrices of an arbitrary dimension $n$, are considered. Additionally it supposed that matrices $P, Q$ and $R$ are not zero matrices and they are not proportional to the unit matrix.