  Let $K$ be any field of characteristic two and let $U_1$ and $W_1$ be the Lie
algebras of the derivations of the algebra of Laurent polynomials $K[t,t^{-1}]$
and of the polynomial ring $K[t]$, respectively. The algebras $U_1$ and $W_1$
are equipped with natural $\mathbb{Z}$-gradings. In this paper, we provide
bases for the graded identities of $U_1$ and $W_1$, and we prove that they do
not admit any finite basis.
