Complete solutions to families of quartic Thue equations

Using a method due to E. Thomas, we prove that if $\vert a\vert > 9.9 \cdot {10^{27}}$ then the Diophantine equations

$${x^4} - a{x^3}y - {x^2}{y^2} + ax{y^3} + {y^4} = 1$$

and

$${x^4} - a{x^3}y - 3{x^2}{y^2} + ax{y^3} + {y^4} = \pm 1$$

have exactly twelve solutions, namely $(x,y) = (0, \pm 1), ( \pm 1,0), ( \pm 1, \pm 1), ( \mp 1, \pm 1), ( \pm a, \pm 1), ( \pm 1, \mp a)$ and eight solutions, $(x,y) = (0, \pm 1), ( \pm 1,0), ( \pm 1, \pm 1), ( \pm 1, \mp 1)$ , respectively.