Summary: The family of second-order linear elliptic operators on the complex plane forms an open set in \(\mathbb{C}^6\) with exactly six components. Let \(E(\Delta)\) denote the component consisting of operators that are deformable to the Laplacian. The objective of this paper is the establishment of the Fredholm alternative for the equation \[ SW(z) = g(z),\;W\in W_0^{2,p} (\Omega),\;g \in L^p (\Omega). \] Using the Hilbert Transforms, the problem reduces to studying \(Lw= g(z)\), where \(L\) is an integral operator and \(\omega \in L^p(\Omega)\). We show that, if the coefficients of \(S\) are functions of vanishing mean oscillation, then \(L\) is a Fredholm operator with index zero for every \(p\in (1,\infty)\).