Always Two

When $b=1$, you can substitute this into the original equations to get a set of two equations in $a$ and $c$.  With a bit of manipulation you can turn these into a quadratic equation in either $a$ or $c$.  Solve this to find some possible values.

When $a=c$ you can rewrite $c$ as $a$ in the original equations, giving you two equations in $a$ and $b$. You can eliminate $b$ to get a cubic equation in $a$. Use the work you have already done in the previous case to find a possible solution for $a$ and use this to factorise the cubic into a product of a linear and quadratic term.

In the video below Claire shows you how you could factorise a cubic.