Simple non-weight modules over Lie superalgebras of Block type

In this paper, a family of non-weight modules over Lie superalgebras $S(q)$ of Block type are studied. Free $U(\eta )$-modules of rank $1$ over Ramond-Block algebras and free $U(\mathfrak{h})$-modules of rank $2$ over Neveu-Schwarz-Block algebras are constructed and classified. Moreover, the sufficient and necessary conditions for such modules to be simple are presented, and their isomorphism classes are also determined. The results cover some existing results.