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In this paper, we define and study the classical   $R$  -matrix for Lie conformal algebras (LCAs), based on which we propose to construct a new LCA. We present a systematic approach for constructing the   $R$  -matrix for an affine Kac–Moody LCA and study the universal vertex algebra associated with the new LCA obtained via the   $R$  -matrix. As an application, using the classical   $R$  -matrix we defined, we introduce a novel integrability scheme to construct infinite-dimensional (Liouville) integrable systems via the Feigin–Frenkel center. A scheme-theoretical explanation of our equations and the case of classical   $\mathscr{W}$  -algebras in our theory will be discussed in subsequent papers.

Generalized AKS Scheme of Integrability via Vertex Algebra