Non-degenerate para-complex structures in 6D with large symmetry groups

For an almost product structure J on a manifold M of dimension 6 with non-degenerate Nijenhuis tensor \(N_J\), we show that the automorphism group \(G=\mathrm{Aut}(M,J)\) has dimension at most 14. In the case of equality G is the exceptional Lie group \(G_2^*\). The next possible symmetry dimension is proved to be equal to 10, and G has Lie algebra \(\mathfrak {sp}(4,{\mathbb R})\). Both maximal and submaximal symmetric structures are globally homogeneous and strictly nearly para-Kähler. We also demonstrate that whenever the symmetry dimension is at least 9, then the automorphism algebra acts locally transitively.