Abel's theorem in problems and solutions - School of Mathematics

242 Appendix by Khovanskii
A.10.3 The integrable case
Let us come back to the problem of the representability of the function
by generalized quadratures.
We consider now the different possible cases and we prove that the
condition we have found is not only necessary but also sufficient for the
representability of the function by generalized quadratures.
FIRST INTEGRABILITY CASE. The group H(G) has an invariant point.
This means that the continuations of the edges of the polygon G intersect
in a point. Sending this point to infinity by a homographic transformation,
we obtain the polygon bounded by segments of straight lines (cf.,
Figure 134).
FIGURE 134
All mappings in have the form All germs of the
function at a non-singular point are obtained by applying to
a given germ the group The germ
is invariant under the action of the group This means that the
germ is the germ of a single-valued function. A singular point of
the function can be only a pole (cf., the Proposition in §A.10.1). Thus
the function is rational. The equation is integrable by
quadratures. This case of integrability is well known. The function in
this case is called the Christoffel–Schwarz integral.
SECOND INTEGRABILITY CASE. The invariant set of the group H(G)
consists of two points. This means that there are two points with the
following properties: for every side of the polygon G these points either