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A new two-mode version of the generalized Zakharov-Kuznetsov equation is derived using Korsunsky’s method. This dynamical model describes the propagation of two-wave solitons moving simultaneously in the same direction with mutual interaction that depends on an embedded phase-velocity parameter. Three different methods are used to obtain exact bell-shaped soliton solutions and singular soliton solutions to the proposed model. Two-dimensional and three-dimensional plots are also provided to illustrate the interaction dynamics of the obtained two-wave exact solutions upon increasing the phase-velocity parameter.