Sistema óptimo, soluciones invariantes y clasificación completa del grupo de simetrías de Lie para la ecuación de Kummer-Schwarz generalizada y su representación del álgebra de Lie

We obtain the complete classification of the Lie symmetry group and the optimal system’s generating operators associated with a particular case of the generalized Kummer - Schwarz equation. Using those operators we characterize all invariant solutions, alternative solutions were found for the equation studied and the Lie algebra associated with the symmetry group is classified.

PROPERTIES OF A DIFFERENTIAL SEQUENCE BASED UPON THE KUMMER-SCHWARZ EQUATION

In this paper, we determine a recursion operator for the Kummer-Schwarz equation, which leads to a sequence with unacceptable singularity properties. A different sequence is devised based upon the relationship between the Kummer-Schwarz equation and the first-order Riccati equation for which a particular generator has been found to give interesting and excellent properties. We examine the elements of this sequence in terms of the usual properties to be investigated – symmetries, singularity properties, integrability, alternate sequence – and provide an explanation of the curious relationship between the results of the singularity analysis and a consideration of the solution of each element obtained by quadratures.

FURTHER GENERALISATIONS OF THE KUMMER-SCHWARZ EQUATION: ALGEBRAIC AND SINGULARITY PROPERTIES

The Kummer–Schwarz Equation, 2<em>y'y'''</em>− 3(<em>y''</em>)² = 0, has a generalisation, (<em>n</em> − 1)<em>y</em>^{(<em>n</em>−2)}<em>y</em>^(<em>n</em>) − <em>ny</em>^{(<em>n</em>−1)²} = 0, which shares many properties with the parent form in terms of symmetry and singularity. All equations of the class are integrable in closed form. Here we introduce a new class, (<em>n</em>+q−2)<em>y</em>^{(<em>n</em>−2})<em>y</em>^(<em>n</em>) −(<em>n</em>+<em>q</em>−1)<em>y</em>^{(<em>n</em>−1)²} = 0, which has different integrability and singularity properties.