Abstract: The main purpose of this article is to construct the vertex algebra of associated to finite-nondegenerate solvable Lie algebra. Avoid the notion of module of vertex algebra and tire some the Jacobi identity. Apply the equivalent condition with the Jacobi identity:the weak commutativity and the D-derivatve-bracket formula. On the theorems of the representation of finite-nondegenerate solvable Lie algebra g and the level l restricted module of the affine algebra $\hat{g}$ of g. Construct and prove a kind the vertex algebras , which equipped the structure of different with that the vertex algebra of associated to Heisenberg algebra and non-twist Kac-moody algebra.

Key words: The level l restricted module of the affine algebra $\hat{g}$, The vertex algebra of associated to finite-nondegenerate solvable Lie algebra, Jacobi-identity and the weak associativity and the D-derivative-bracket formulas