Essentially generalizing Lie’s results, we prove that the contact equivalence groupoid of a class of (1+1)-dimensional generalized nonlinear Klein–Gordon equations is the first-order prolongation of its point equivalence groupoid, and then we carry out the complete group classification of this class. Since it is normalized, the algebraic method of group classification is naturally applied here. Using the specific structure of the equivalence group of the class, we essentially employ the classical Lie theorem on realizations of Lie algebras by vector fields on the line. This approach allows us to enhance previous results on Lie symmetries of equations from the class and substantially simplify the proof. After finding a number of integer characteristics of cases of Lie-symmetry extensions that are invariant under action of the equivalence group of the class under study, we exhaustively describe successive Lie-symmetry extensions within this class.

本质上推广Lie的结果，证明一类(1+1)维广义非线性Klein-Gordon方程的接触等价群是其点等价群的一阶延拓，然后进行完全群分类这个班的。既然是归一化的，这里自然就用到了群分类的代数方法。利用类的等价群的具体结构，我们实质上采用了经典的李定理在线向量场实现李代数。这种方法使我们能够增强先前关于该类方程的李对称性的结果，并大大简化了证明。