Given an indirected graph G = (V, E), a Vertex k-Separator is a subset of the vertex set V such that, when the separator is removed from the graph, the remaining vertices can be partitioned into k subsets that are pairwise edge-disconnected. In this paper we focus on the Balanced Vertex k-Separator Problem, i.e., the problem of finding a minimum cardinality separator such that the sizes of the resulting disconnected subsets are balanced. We present a compact Integer Linear Programming formulation for the problem, and present a polyhedral study of the associated polytope. We also present an Exponential-Size formulation, for which we derive a column generation and a branching scheme. Preliminary computational results are reported comparing the performance of the two formulations on a set of benchmark instances.