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In this article, we focus on investigating the new coupled Konno–Oono equation that arises in the magnetic field. An effective technology called the Exp-function method (EFM) is utilized to find abundant analytical solutions. By this method, four families (28 sets) of the exact solutions, such as bright solitary, dark solitary, bright–dark solitary, double-bright solitary, double-dark solitary and kinky bright–dark solitary wave solutions, are constructed. The performances of the real, imaginary and absolute parts of the solutions are presented in the form of 3D contours. The results show that the EFM is a promising method to construct abundant analytical solutions for the partial differential equations arising in physics.