Let's say I have an elliptic differential $R(t,\sqrt{f(t)})$, where $f(t)$ is a fourth or third order polynomial. I want to prove it can be transformed by a Möbius transform $t\rightarrow\frac{at+b}{ct+d}$ into a form for which either $f(t)=t(t-1)(t-\lambda)$, $f(t)=t^3+at+b$ or$f(t)=(1-t^2)(1-k^2t^2)$).
EDIT: Reformulated question after remarks.