On partially hyperbolic symplectic automorphisms of a 6-torus

We study automorphisms of a $6$-torus with one-dimensional stable and unstable manifolds, and a four-dimensional center manifold. Such automorphisms are generated by integer matrices and are symplectic with respect to either the standard symplectic structure on $\mathbb{R}^6$ or a nonstandard symplectic structure generated by an integer skew-symmetric nondegenerate matrix. Such a symplectic matrix generates a partially hyperbolic automorphism of the torus if its eigenvalues are given by a pair of real numbers outside the unit circle and two pairs of conjugate complex numbers on the unit circle. The classification is determined by the topology of a foliation generated by unstable (stable) leaves of the automorphism and its action on the center manifold. There are two different cases, transitive and decomposable ones. In the first case, the foliation into unstable (stable) leaves is transitive, and in the second case, the foliation itself has a subfoliation into $2$-dimensional or $4$-dimensional tori.