Third order differential equations with fixed critical points (Q5901458)

Fuchs determined all algebraic differential equations of first order whose solutions have no movable singularities. For the second order equation \(y''=F(z,y,y')\) this was done by Painlevé and his school. Third order equations were considered by Chazy, Garnier and Bureau. The present paper deals with the specific third order equation \[ \begin{multlined} y'''(\delta y'+\alpha y^2) =\beta {y''}^2+a_1yy'y''+a_2y^3y'' +a_3{y'}^3 +a_4y^2{y'}^2 +a_5y^4y'\\ +a_6y^6 +a_7y^2y'' +a_8y{y'}^2 +a_9y^4+a_{10}y^2y' +a_{11}y^3.\end{multlined} \]