The paper investigates Calogero-Degasperis-Fokas (CDF) equation, an exactly solvable third order nonlinear evolution equation (Fokas, 1980). All possible functions for the unknown function $F(\nu )$ in the considered equation are listed that contains the nontrivial Lie point symmetries. Furthermore, nonlinear self-adjointness is considered and for the physical parameter $A\ne 0$ the equation is proved not strictly self-adjoint equation but it is quasi self-adjoint or more generally nonlinear self-adjoint equation. In addition, it is remarked that CDF equation admits a minimal set of Lie algebra under invariance test of Lie groups. Subsequently, Lie symmetry reductions of CDF equation are described with the assistance of an optimal system, which reduces the CDF equation into different ordinary differential equations. Besides, Lie symmetries are used to indicate the associated conservation laws. Also, the well-known $(G^{\prime }/G)$-expansion approach is applied to obtain the exact solutions. These new periodic and solitary wave solutions are feasible to analyse many compound physical phenomena in the field of sciences.

本文研究了Calogero-Degasperis-Fokas（CDF）方程，这是一个完全可解的三阶非线性发展方程（Fokas，1980）。未知功能的所有可能功能$F（\nu ）$在所考虑的方程中列出了包含非平凡Lie点对称性的参数。此外，考虑了非线性自伴和物理参数$\mathrm{一种}\ne 0$证明该方程不是严格的自伴方程，而是准自伴或更一般的非线性自伴方程。另外，值得注意的是，在李群不变性检验下，CDF方程允许最小的李代数集。随后，借助最优系统描述了CDF方程的Lie对称约简，该方程将CDF方程简化为不同的常微分方程。此外，李对称性用来表示相关的守恒定律。还有，著名的$（G^{\prime }/G）$-扩展方法用于获得精确解。这些新的周期波和孤波解对于分析科学领域的许多复合物理现象是可行的。