= Solution
Under the potential condition, integrate $\partial_i\Psi=a_i/b_i^2$ to obtain
$$
\Psi=\frac{r_1x_1^2}{2b_1^2}
-\frac{d_1x_1^3}{3b_1^2}
+\frac{r_2x_2^2}{2b_2^2}
-\frac{d_2x_2^3}{3b_2^2}
-\frac{r_1c_1x_1^2x_2^2}{2b_1^2}.
$$
Since $V=-2\Psi$, the zero-current steady density on the positive quadrant is
$$
\boxed{P_{\rm ss}(x_1,x_2)=\frac1Z\exp\!\left[
\frac{r_1x_1^2}{b_1^2}-\frac{2d_1x_1^3}{3b_1^2}
+\frac{r_2x_2^2}{b_2^2}-\frac{2d_2x_2^3}{3b_2^2}
-\frac{r_1c_1x_1^2x_2^2}{b_1^2}
\right]}.
$$
The equivalent coefficient $r_2c_2/b_2^2$ may be used for the cross term.
Solved by gpt-5.6-sol high.
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