Verma modules for rank two Heisenberg-Virasoro algebra

Let $⪯$ be a compatible total order on the additive group ${\mathbb{Z}}^2$, and $L$ be the rank two Heisenberg-Virasoro algebra. For any $\mathbf{c}=(c_1,c_2,c_3,c_4)\in {\mathbb{C}}^4$, we define ${\mathbb{Z}}^2$-graded Verma module $M(\mathbf{c},⪯)$ for the Lie algebra $L$. A necessary and sufficient condition for the Verma module $M($\mathbf{c}$, \preceq)$tobeirreducibleisprovided\mathrm{.}Moreover,themaximal$$\mathbb{Z}$^2$-gradedsubmodulesoftheVermamodule$M($\mathbf{c}$, \preceq)$arecharacterizedwhen$M($\mathbf{c}$, \preceq)$ is reducible.