Any nonlinear equation of the form y''=\sum_{n=0}^N a_n(z)y^n
has a (generally branched) solution with leading order behaviour
proportional to
(z-z_0)^{-2/(N-1)} about a point z_0, where the coefficients a_n are analytic at z_0 and a_N(z_0)\ne 0. Jointly with R.G. Halburd we consider the subclass of equations for which each possible leading order term of
this
form corresponds to a one-parameter family of solutions represented near
z_0
by a Laurent series in fractional powers of z-z_0. For this class of
equations we show that the only movable singularities that can be reached
by
analytic continuation along finite-length curves are of the algebraic type
just described. This work generalizes previous results of S. Shimomura.
The only other possible kind of movable singularity that might occur is an
accumulation point of algebraic singularities that can be reached by
analytic continuation along infinitely long paths ending at a finite point
in the complex plane. This behaviour cannot occur for constant coefficient
equations in the class considered. However, an example of R. A. Smith
shows
that such singularities do occur in solutions of a simple autonomous
second-order differential equation outside the class we consider here.