  In this paper, we define and study the classical $R$-matrix for vertex Lie
algebra, based on which we propose to construct a new vertex Lie algebra. We
give a systematic way to construct the $R$-matrix for affine Kac-Moody vertex
Lie algebra and study the universal vertex algebra associated with the new
vertex Lie algebra that we obtained by $R$-matrix. As an application, using the
classical $R$-matrix we defined, we give a new scheme to construct
infinite-dimensional (Liouville) integrable systems via the Feigin-Frenkel
center. The scheme-theoretical explanation of our equations and the classical
$\mathcal{W}$-algebra case of our theory will come later.
