A case of the generalized vanishing conjecture

In this paper, we show that the GVC (generalized vanishing conjecture) holds for the differential operator $\mathrm{\Lambda }=({\mathrm{\partial }}_x-\mathrm{\Phi }({\mathrm{\partial }}_y)){\mathrm{\partial }}_y$ and all polynomials $P(x,y)$, where $\mathrm{\Phi }(t)$ is any polynomial over the base field. The GVC arose from the study of the Jacobian conjecture.