  This work introduces a new class of $F$-structures satisfying $\alpha F^{K+1}
+\beta F^{K} + F=0$, where $K$ is a positive integer, $K\geq3$, and $\alpha,
\beta$ are real or complex numbers. We investigate the Cauchy-Riemann structure
and its link with the $F$-structure. We also address the integrability of this
structure, including partial and complete integrability. Finally, we present
several examples of the $F$-structure.
