The study of invariant states of fermionic lattice systems begun earlier is contined. Under the assumption that the time dynamics corresponds to a (formal) Hamiltonian $H_0$ and the invariant state $\varphi$ is a KMS state for some Hamiltonian $H$ [1], one-dimensional lattice Fermi systems were considered in the earlier work. In particular, the case when $H_0$ is not a quadratic form in the creation and annihilation operators and all nonquadratic terms in $H_0$ are diagonal was studied. In this case, it was shown that up to an arbitrary diagonal quadratic form $N$ the Hamiltonian $H$ is proportional to $H_0$, i. e., that $\varphi$ is a KMS state of $\beta H_0+ N$. In this paper, we obtain a similar result for Fermi systems of arbitrary dimension by a somewhat different method to the one used earlier [1].