Extended affine Lie algebras, vertex algebras and equivariant $\phi$-coordinated quasi modules

For any nullity $2$ extended affine Lie algebra $\mathcal{E}$ of maximal type and $\ell\in\mathbb{C}$, we prove that there exist a vertex algebra $V_{\mathcal{E}}(\ell)$ and an automorphism group $G$ of $V_{\mathcal{E}}(\ell)$ equipped with a linear character $\chi$, such that the category of restricted $\mathcal{E}$-modules of level $\ell$ is canonically isomorphic to the category of $(G,\chi)$-equivariant $\phi$-coordinated quasi $V_{\mathcal{E}}(\ell)$-modules. Moreover, when $\ell$ is a nonnegative integer, there is a quotient vertex algebra $L_{\mathcal{E}}(\ell)$ of $V_{\mathcal{E}}(\ell)$ modulo by a $G$-stable ideal, and we prove that the integrable restricted $\mathcal{E}$-modules of level $\ell$ are exactly the $(G,\chi)$-equivariant $\phi$-coordinated quasi $L_{\mathcal{E}}(\ell)$-modules.