High integer spins beyond the Fierz-Pauli Framework

We explicitly construct in the (1/2,1/2)X(1/2,1/2) representation space the operator of the squared Pauli-Lubanski vector and derive from it that the -6m^{2}eigensubspace (spin 2 in the rest frame), with well defined parity, is pinned down by the one sole equation, [\epsilon_{\mu\alpha\beta\sigma} \epsilon^{\mu}_{\nu\delta\rho}p^{\sigma}p^{\rho}-m^{2}g_{\alpha\beta}g_{\nu\del ta}]h^{\beta\delta}=0$.