  This article describes the use of algebraic methods in a phase plane analysis
of ordinary differential equations. The method is illustrated by the study of
capillary-gravity steady surface waves propagating in shallow water. We
consider the (fully nonlinear, weakly dispersive) Serre-Green-Naghdi equations
with surface tension, because it provides a tractable model that, in the same
time, is not too simple so the interest of the method can be emphasised. In
particular, we analyse a special class of solutions, the solitary waves, which
play an important role in many fields of Physics. In capillary-gravity regime,
there are two kinds of localised infinitely smooth travelling wave solutions --
solitary waves of elevation and of depression. However, if we allow the
solitary waves to have an angular point, the "zoology" of solutions becomes
much richer and the main goal of this study is to provide a complete
classification of such singular localised solutions using the methods of the
effective Algebraic Geometry.
