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 $G=Aut(M,J)$ 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 $sp(4,R)$. Both maximal and submaximal symmetric structures are globally homogeneous and strictly nearly para-K\"ahler. We also demonstrate that whenever the symmetry dimension is at least 9, then the automorphism algebra acts locally transitively.