We study $G$-almost geodesic mappings of the second type $\underset{\theta }{\to }{\pi }_2\left(e\right)$, $\theta =1,2$ between non-symmetric affine connection spaces. These mappings are a generalization of the second type almost geodesic mappings defined by N. S. Sinyukov (1979). We investigate a special type of these mappings in this paper. We also consider $e$-structures that generate mappings of type $\underset{\theta }{\to }{\pi }_2\left(e\right)$, $\theta =1,2$. For a mapping $\underset{\theta }{\to }{\pi }_2(e,F)$, $\theta =1,2$, we determine the basic equations which generate them.

We define the concept of a bi-Legendrian connection associated to a bi-Legendrian structure on an almost -manifold $M^{2n+r}$. Among other things, we compute the torsion of this connection and prove that the curvature vanishes along the leaves of the bi-Legendrian structure. Moreover, we prove that if the bi-Legendrian connection is flat, then the bi-Legendrian structure is locally equivalent to the standard structure on ${ℝ}^{2n+r}$.