Singularities of the First Painlev{é} Transcendent

Consider the solution $y(t)$ for the ordinary differential equation $y^{\mathrm{\prime }}=f(t,y)$ with $t$ complex. Second-order nonlinear differential equations often exhibit patterns in their poles, branch points, and essential singularities, explored by \Pain and colleagues, 1888–1915. A variant of the ratio test applied to the Taylor series for the solution $y$ estimates the locations and orders of singularities in the First Painlev{é} Transcendent as an example. Can you suggest applications in which our singularity location analysis can provide useful insights?