For a given skew symmetric real n x n matrix N, the bracket X, Y = XNY - YNX defines a Lie algebra structure on the space Sym(n, N) of symmetric n x n real matrices and hence a corresponding Lie-Poisson structure. The purpose of this paper is to investigate the geometry, integrability, and linearizability of the Hamiltonian system. (X) over dot = [X-2, N], or equivalently in Lax form, the equation. (X) over dot = [X, XN + NX] on this space along with a detailed study of the Poisson geometry itself. If N has distinct eigenvalues, it is proved that this system is integrable on a generic symplectic leaf of the Lie-Poisson structure of Sym(n, N). This is established by finding another compatible Poisson structure.