  The core object of this paper is a pair $(L, e)$, where $L$ is a Nijenhuis
operator and $e$ is a vector field satisfying a specific Lie derivative
condition, i.e., $Lie_{e}L=\operatorname{Id}$. Our research unfolds in two
parts. In the first part, we establish a Splitting Theorem for Nijenhuis
operators with a unity, offering an effective reduction of their study to cases
where $L$ has either one real or two complex conjugate eigenvalues at a given
point. We further provide the normal forms for $\mathrm{gl}$-regular Nijenhuis
operators with a unity around algebraically generic points, along with
semi-normal forms for dimensions two and three. In the second part, we
establish the relationship between Nijenhuis operators with a unity and
$F$-manifolds. Specifically, we prove that the class of regular $F$-manifolds
coincides with the class of Nijenhuis manifolds with a cyclic unity. By
extending our results from dimension three, we reveal semi-normal forms for
corresponding $F$-manifolds around singularities.
