$\mathbb{Z}$-graded identities of the Lie algebras $U_1$

Let $K$ be an infinite field of characteristic different from two and let $U_1$ be the Lie algebra of the derivations of the algebra of Laurent polynomials $K[t,t^{-1}]$. The algebra $U_1$ admits a natural $\mathbb{Z}$-grading. We provide a basis for the graded identities of $U_1$ and prove that they do not admit any finite basis. Moreover, we provide a basis for the identities of certain graded Lie algebras with a grading such that every homogeneous component has dimension $\leq 1$, if a basis of the multilinear graded identities is known. As a consequence of this latter result we are able to provide a basis of the graded identities of the Lie algebra $W_1$ of the derivations of the polynomial ring $K[t]$. The $\mathbb{Z}$-graded identities for $W_1$, in characteristic 0, were described in \cite{FKK}. As a consequence of our results, we give an alternative proof of the main result, Theorem 1, in \cite{FKK}, and generalize it to positive characteristic. We also describe a basis of the graded identities for the special linear Lie algebra $sl_q(K)$ with the Pauli gradings where $q$ is a prime number.