  In this paper, we consider the homogenization problem for generalized
elliptic systems $$
\mathcal{L}_{\varepsilon}=-\operatorname{div}(A(x/\varepsilon)\nabla+V(x/\varepsilon))+B(x/\varepsilon)\nabla+c(x/\varepsilon)+\lambda
I $$ with dimension two. Precisely, we will establish the $ W^{1,p} $
estimates, H\"{o}lder estimates, Lipschitz estimates and $ L^p $ convergence
results for $ \mathcal{L}_{\varepsilon} $ with dimension two. The operator $
\mathcal{L}_{\varepsilon} $ has been studied by Qiang Xu with dimension $ d\geq
3 $ in \cite{Xu1,Xu2} and the case $ d=2 $ is remained unsolved. As a
byproduct, we will construct the Green functions for $
\mathcal{L}_{\varepsilon} $ with $ d=2 $ and their convergence rates.
