Some modules over Lie algebras related to the Virasoro algebra

I will talk about  restricted modules over  a class of $\frac12\mathbb Z$-graded Lie algebras  $\mathfrak g$ related to the Virasoro  algebra.  We in fact give the classification of certain  irreducible restricted $\mathfrak g$-modules in the sense of determining each irreducible restricted module up to an irreducible module over a subalgebra of $\mathfrak g$ which contains its positive part.   By the correspondence between restricted modules over $\mathfrak g$  and modules over the vertex algebra associated to $\mathfrak g$,  we  get  the classification of  certain  irreducible modules over  vertex algebras associated to these $\mathfrak g.$