  In this paper, we study representations of non-finitely graded Lie algebras
$\mathcal{W}(\epsilon)$ related to Virasoro algebra, where $\epsilon = \pm 1$.
Precisely speaking, we completely classify the free $\mathcal{U}(\mathfrak
h)$-modules of rank one over $\mathcal{W}(\epsilon)$,and find that these module
structures are rather different from those of other graded Lie algebras. We
also determine the simplicity and isomorphism classes of these modules.
