Let K be an infinite field of characteristic different from two and let U1 be the Lie algebra of the derivations of the algebra of Laurent polynomials Kt,t−1. The algebra U1 admits a natural Z-grading. We provide a basis for the graded identities of U1 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 ≤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 W1 of the derivations of the polynomial ring Kt. The Z-graded identities for W1, in characteristic 0, were described in 8. As a consequence of our results, we give an alternative proof of the main result, Theorem 1, in 8, and generalize it to positive characteristic. We also describe a basis of the graded identities for the special linear Lie algebra slq(K) with the Pauli gradings where q is a prime number.