We consider a deformed fifth-order Korteweg–de Vries (D5oKdV) equation and investigated its integrability and group theoretical aspects. By extending the well-known Lax pair technique, we show that the D5oKdV equation admits a Lax representation provided that the deformed function satisfies certain differential constraint. It is observed that the D5oKdV equation admits the same differential constraint (on the deforming function) as that of the deformed Korteweg–de Vries (DKdV) equation. Using the Lax representation, we show that the D5oKdV equation admits infinitely many conservation laws, which guarantee its integrability. Finally, we apply the Lie symmetry analysis to the D5oKdV equation and derive its Lie point symmetries, the associated similarity reductions and the exact solutions.

中文翻译：

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变形五阶 Korteweg-de Vries 方程的可积性和精确解

我们考虑变形的五阶 Korteweg-de Vries (D5oKdV) 方程并研究其可积性和群论方面。通过扩展众所周知的 Lax 对技术，我们表明 D5oKdV 方程承认 Lax 表示，前提是变形函数满足某些微分约束。据观察，D5oKdV 方程与变形的 Korteweg-de Vries (DKdV) 方程具有相同的微分约束（对变形函数）。使用 Lax 表示，我们表明 D5oKdV 方程承认无限多个守恒定律，这保证了其可积性。最后，我们将李对称性分析应用于 D5oKdV 方程并推导出其李点对称性、相关的相似度减少和精确解。