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The aim of this paper is to deal with the anisotropic doubly criticalequation $$-\Delta_p^H u - \frac{\gamma}{[H^\circ(x)]^p} u^{p-1} = u^{p^*-1}\qquad \text{in } \R^N,$$ where $H$ is in some cases called Finsler norm,$H^\circ$ is the dual norm, $1<p<N$, $0 \leq \gamma< \left((N-p)/p\right)^p$and $p^*=Np/(N-p)$. In particular, we provide a complete  asymptotic analysis of $u \in \mathcal{D}^{1,p}(\R^N)$ near the origin and atinfinity, showing that this solution has the same features of its euclideancounterpart. Some of the techniques used in the proofs are new even in theEuclidean framework.

Asymptotic behaviour of solutions to the anisotropic doubly critical equation