Computing relative power integral bases in a family of quartic extensions of imaginary quadratic fields

Let $M=\mathbb{Q}(\sqrt{-D})$ be an imaginary quadratic field with the ring of integers $\mathbb{Z}_{M}$, and let $\xi$ be a root of the polynomial $f(x)=x^{4}-2cx^{3}+2x^{2}+2cx+1$, where $c\in\mathbb{Z}_{M}\setminus\left\{0,\pm2\right\}$ and $c\neq\pm1$ if $D=1$ or $3$. We consider an infinite family of octic fields $K_{c}=M\left(\xi\right)$ with the ring of integers $\mathbb{Z}_{K_{c}}$. Our goal is to determine all generators of a relative power integral basis of $\mathcal{O}=\mathbb{Z}_{M}\left[\xi\right]$ over $\mathbb{Z}_{M}$. We show that our problem reduces to solving the system of relative Pellian equations $cV^{2}-\left(c+2\right)U^{2}=-2\mu$, $cZ^{2}-\left(c-2\right)U^{2}=2\mu$, where $\mu$ is a unit in $\mathbb{Z}_{M}$. We solve the system completely and find that all non-equivalent generators of power integral bases of $\mathcal{O}$ over $\mathbb{Z}_{M}$ are given by $\alpha=\xi$, $2\xi-2c\xi^{2}+\xi^{3}$ for $\left\vert c\right\vert\geq159108$ and $|c|\leq1000$, $c\notin S_{c}$ (where $S_{c}$ is a set of exceptional cases, $|S_{c}|=28$). Also, we find that, in all the above cases, $\mathcal{O}$ admits no absolute power integral basis if $-D\equiv2,3(\operatorname{mod}4)$.