  In this paper, we classify all positive solutions for the following integral
equation: \begin{equation} u(x)=\int_{\mathbb{R}^n_+}K_b(x,y)y_n^b f(u(y))dy,
\end{equation} where $ b > 1$ is a constant. Here $ K_b(x,y)$ is the Green
function of the following homogeneous Neumann boundary problem \begin{equation}
\left\{ \begin{aligned} -\text{div}(x^{b}_n \nabla u)&= f \quad in
\mathbb{R}^n_+ \\ \frac{\partial u}{\partial x_n}&= 0 \quad on \ \partial
\mathbb{R}^n_+ . \end{aligned} \right. \end{equation} By using the method of
moving planes in integral form, we derive the symmetry of positive solutions.
We also establish the equivalence between the integral equation and its
corresponding partial differential equation. Similarly, the results can be
generalized to the integral system.
