In this paper, we define a new variant of the knapsack problem, the Symmetric Weight Matrix Knapsack Problem (SMKP). The (SMKP) is the core structure of the Hydro Unit Commitment being a production scheduling problem relative to hydroelectric units. The (SMKP) is proven to be NP-hard. The main contribution is a polyedral study of the (SMKP) and a dedicated two-phase Branch & Cut scheme. The polyedral study focuses on inequalities with 0-1 coefficients. Necessary facet-defining conditions are described, through a new structure called pattern encoding the symmetries of the (SMKP). A special case of patterns is identified where these conditions are necessary and sufficient. A two-phase Branch & Cut scheme is defined to exploit the symmetries on the pattern inequalities. As a pre-processing step, the first phase aims to produce patterns verifying these conditions. The second phase separates the inequalities associated to the patterns produced in the first phase at the nodes of a Branch & Cut tree. Experimental results demonstrate the efficiency of the proposed scheme in particular with respect to default CPLEX.