Invariance of an almost contact metric structure of a smooth manifold with respect to the characteristic vector

In this paper, we obtain criteria for the $\Phi$-invariance and the $\eta$-invariance for almost contact metric structures and a criterion for a characteristic vector $\xi$ to be a Killing vector. We find all classes of almost contact metric structures from the Kirichenko classification that are $\Phi$-invariant, $\eta$-invariant, and $\xi$ is a Killing vector. Also, we prove that for any almost contact metric structure, $\xi$ cannot be conformal Killing vector distinct from a Killing vector.