  Given $N\geq 2$, we completely classify the solutions of the anisotropic
$N$-Liouville equation $$-\Delta_N^H\,u=e^u \quad\text{in }\mathbb{R}^N,$$
under the finite mass condition $\int_{\mathbb{R}^N} e^u\,dx<+\infty$. Here
$\Delta_N^H$ is the so-called Finsler $N$-Laplacian induced by a positively
homogeneous function $H$. As a consequence for $N=2$, we give an affirmative
answer to a conjecture made in [G. Wang and C. Xia, J. Differential Equations
252 (2012) 1668--1700].
