On sufficient conditions for the existence of solutions for first order equations and fourth degree with the Painlevé property (Q290294)

The authors consider the ordinary differential equation \[ (w')^4+P(z,w)(w')^3+Q(z,w)(w')^2+R(z,w)w'+S(z,w)=0, \] where \(P(z,w)\), \(Q(z,w)\), \(R(z,w) \) and \(S(z,w)\) are polynomials with respect to \(w\) with analytical coefficients in \(z\). They give explicit sufficient conditions on the coefficients \(P(z,w),Q(z,w),R(z,w) \) and \(S(z,w)\) for this equation to have fixed critical points. Also the following list of Fuchsian differential equations is given \[ w'=a_0+a_1w^2 \] where \(a_0\) and \(a_1\) are analytical functions. \[ \begin{aligned} &(w')^2=c_0+c_1w+c_3w^3,\quad c_i\in\mathbb C,c_3\neq 0,\\ &(w')^2=c_0+c_1w+c_2w^2+c_4w^4, \quad c_i\in\mathbb C,\\ &(w'+\frac{3}{8}(c_0+c_2w^2))^3(w'-\frac{1}{8}(c_0+c_2w^2))=\gamma,\quad c_i,\gamma\in\mathbb C,c_2\neq 0. \end{aligned} \]