Exploring the Cauchy-Riemann structure and integrability conditions of $F$-structures satisfying $\alpha F^{K+1}+\beta F^K+F=0$

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, $K\ge 3$, 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.