Analysis of modified Painlevé-Ince equations (Q2147948)

This paper is concerned with the modified Painlevé-Ince equation \[ y^{\prime\prime}+\alpha yy^{\prime} +\beta y^{3} =0\tag{1} \] and the generalised Painlevé-Ince equations \[ y^{\prime\prime}+\alpha yy^{\prime} +\beta y^{3}+\lambda_1^{}y =0,\tag{2} \] \[ y^{\prime\prime}+\alpha yy^{\prime} +\beta y^{3}+\lambda_1^{}y+\lambda_2^{} =0. \] Using Lie symmetries, Noether symmetries and factorization method, the author provides a comprehensive analysis of these ordinary differential equations. Also a good overview of the existing results of integrability for these equations is presented.