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<p>Lie symmetries play a central role in the analysis and solution of differential equations, offering systematic techniques for order reduction and integration. However, many differential equations do not admit classical Lie point symmetries, limiting the applicability of this approach. To address this limitation, various generalisations have been developed, among which the notion of <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\lambda\)</EquationSource> </InlineEquation>-symmetries (also known as <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(C^{\infty }\)</EquationSource> </InlineEquation>-symmetries) has proven particularly effective. These <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\lambda\)</EquationSource> </InlineEquation>-symmetries extend the symmetry framework, enabling the reduction of order and the derivation of first integrals even in the absence of classical symmetries. In this paper, we investigate the Painlevé–Ince equation within the context of <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\lambda\)</EquationSource> </InlineEquation>-symmetries for the first time. We identify the conditions under which the equation admits such symmetries and employ them to obtain a novel reduction of order. This analysis yields new insights into the structural properties and integrability of the equation.</p>

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\(\lambda\)-Symmetries of the Painlevé–Ince equation

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  • 发表日期：

    2025年9月8日

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  • Slungile Tshibase,
  • Keshlan S. Govinder

摘要

Lie symmetries play a central role in the analysis and solution of differential equations, offering systematic techniques for order reduction and integration. However, many differential equations do not admit classical Lie point symmetries, limiting the applicability of this approach. To address this limitation, various generalisations have been developed, among which the notion of \(\lambda\) -symmetries (also known as \(C^{\infty }\) -symmetries) has proven particularly effective. These \(\lambda\) -symmetries extend the symmetry framework, enabling the reduction of order and the derivation of first integrals even in the absence of classical symmetries. In this paper, we investigate the Painlevé–Ince equation within the context of \(\lambda\) -symmetries for the first time. We identify the conditions under which the equation admits such symmetries and employ them to obtain a novel reduction of order. This analysis yields new insights into the structural properties and integrability of the equation.