  We consider the anisotropic Finsler $N$-Laplacian Liouville equation
\[-\Delta ^{H}_{N}u=e^u \qquad {\rm{in}}\,\, \mathcal{C},\] where $N\geq2$,
$\mathcal{C}\subseteq\mathbb{R}^{N}$ is an open convex cone including
$\mathbb{R}^{N}$, the half space $\mathbb{R}^{N}_{+}$ and
$\frac{1}{2^{m}}$-space $\mathbb{R}^{N}_{2^{-m}}:=\{x\in\mathbb{R}^{N}\mid
x_{1},\cdots,x_{m}>0\}$ ($m=1,\cdots,N$), and the anisotropic Finsler
$N$-Laplacian $\Delta ^{H}_{N}$ is induced by a positively homogeneous function
$H(x)$ of degree $1$. All solutions to the Finsler $N$-Laplacian Liouville
equation with finite mass are completely classified. In particular, if
$H(\xi)=|\xi|$, then the Finsler $N$-Laplacian $\Delta ^{H}_{N}$ reduces to the
regular $N$-Laplacian $\Delta_N$. Our result is a counterpart in the limiting
case $p=N$ of the classification results in \cite{CFR} for the critical
anisotropic $p$-Laplacian equations with $1<p<N$ in convex cones, and also
extends the classification results in \cite{CK,CL,CW,CL2,E} for Liouville
equation in the whole space $\mathbb{R}^{N}$ to general convex cones. In our
proof, besides exploiting the anisotropic isoperimetric inequality inside
convex cones, we have also proved and applied the radial Poincar\'{e} type
inequality (Lemma \ref{A1}), which are key ingredients in the proof and of
their own importance and interests.
