Non-weight modules over generalized Heisenberg-Virasoro algebra of rank two

In this paper, we study a class of non-weight modules over the generalized Heisenberg-Virasoro algebra of rank two $\tilde{L}(p_1,p_2)$. We construct a family of irreducible $\tilde{L}(p_1,p_2)$-modules, determine the isomorphism classes and show that these modules exhaust all the $\tilde{L}(p_1,p_2)$-modules that are free modules of rank one over the Cartan subalgebra.