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This paper provides the general traveling wave solution to the generalized  Calogero–Bogoyavlenskii–Schiff (gGCBs) equation, thus allowing the exact analysis of associated nonlinear wave propagation in different physical fields, i.e., soliton theory, fluid dynamics, and optics. First, a traveling wave transformation reduces the said gGCBs partial differential equation into a fourth-order ordinary one that can be readily integrated once, thus yielding a one-parameter family of third-order equations. Using the Lie symmetry method, one performs a two-dimensional abelian Lie symmetry algebra for such a family. The two symmetry generators from the algebra provide differential invariants that map the third-order into a second-order ordinary equation, the so-called auxiliary equation. This auxiliary equation inherits the algebra generators, providing each a first integral. The two first integrals constitute the general solution to the gCBS equation. This solution is given in parametric form once the associated primitives are usually too complicated to be explicitly computed. Explicit forms from the parametric solution arise when the analytical assessment of the primitives is feasible, leading to particular cases. In this framework,  the proposed solution encompasses particular literature solutions.