Complete solutions of a family of quartic Thue and index form equations

Continuing the recent work of the second author, we prove that the diophantine equation

for $|a|\ge 3$ has exactly 12 solutions except when $|a|=4$, when it has 16 solutions. If $\alpha=\alpha(a)$ denotes one of the zeros of $f_a(x,1)$, then for $|a|\ge 4$ we also find all $\gamma\in\mathbb Z[\alpha]$ with $\mathbb Z[\gamma]=\mathbb Z[\alpha]$.