The Nijenhuis tensor of a pseudo-cosymplectic manifold

In this paper, we examine the Nijenhuis tensor of pseudo-cosymplectic manifolds. The adjoint $G$-structure of an almost contact metric manifold is constructed, the first group of such manifolds is defined. The pseudo-cosymplectic subclass of quasi-cosymplectic manifolds is distinguished and the first group of structure equations is obtained for them. We obtained necessary and sufficient conditions under which a pseudo-cosymplectic manifold is cosymplectic, most precisely cosymplectic, normal, or integrable.