Two classes of Lie conformal superalgebras related to the Heisenberg--Virasoro Lie conformal algebra

In this paper, firstly we construct two classes of Lie conformal superalgebras denoted by $\mathcal{H}\mathcal{V}\mathcal{S}(\alpha )$ and $\mathcal{H}\mathcal{V}\mathcal{S}(\beta ,\gamma ,\tau )$, respectively, where $\alpha$ is an nonzero complex number and $\beta ,\gamma ,\tau$ are complex numbers. They are both of rank $(2+1)$ and the even part $\mathcal{H}\mathcal{V}\mathcal{S}(\alpha {)}_{\bar{0}}=\mathcal{H}\mathcal{V}\mathcal{S}(\beta ,\gamma ,\tau {)}_{\bar{0}}$ is the Heisenberg–Virasoro Lie conformal algebra, which is a free $\mathbb{C}[\mathrm{\partial }]$-module generated by $L$ and $H$ satisfying $[L_{\lambda }L]=(\mathrm{\partial }+2\lambda )L,[L_{\lambda }H]=(\mathrm{\partial }+\lambda )H,[H_{\lambda }H]=0$. Then we completely determine the conformal derivations, conformal biderivations, automorphism groups and conformal modules of rank $(1+1)$ for these two Lie conformal superalgebras.