Any nonlinear equation of the form у"=Σ~N_(n=0)α_n(Z)y~n has a solution with leading behavior proportional to (z-z_0)~(-2/(N-1)) about a point z_0, where the coefficients a_n are analytic at Z_0 and aN(z_0)≠0. Equations are considered for which each possible leading term of this form extends to a Laurent series solution in fractional powers of z-z_0. For these 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 generalizes results of Shimomura ["On second order nonlinear differential equations with the quasi-Painlevé property I1," RIMS Kokyuroku 1424, 177 (2005)]. The possibility that these algebraic singularities could accumulate along infinitely long paths ending at a finite point is considered. Smith ["On the singularities in the complex plane of the solutions of y"+y'f(y) +g(y) =P(χ)," Proc. Lond. Math. Soc. 3, 498 (1953)] showed that such singularities do occur in solutions of a simple equation outside this class.
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