It is a well known fact that any second order Fuchsian differential equation on the complex plane $$u''(x) + p(x)u'(x) + q(x)u(x)=0$$ with exactly $4$ regular singular points may be suitably transformed into the canonical form of the Heun equation. Surely, giving this transformation in explicit terms must be possible. However, I have not found a source that actually constructs such a transformation, or perhaps provides an abstract proof of the existence of it (if that is at all possible).

I am interested in the *explicit form* of such a transformation - i.e. a recipe on how to obtain it for a specific given equation.

What I do know: applying a Möbius transformation to the independent variable allows to shift the $4$ regular singular points to the points $0,1,a,\infty$ where $a \in \mathbb{C}$, which is certainly necessary since these are the singular points of the canonical form of the Heun equation. Now I assume one will want to use a transformation of the dependent variable $u$ of the type $u \mapsto (x-A)^{\alpha} (x-B)^{\beta} (x-C)^{\gamma} u(x)$ or something similar. Any suggestions or references?