In this paper, we present an explicit and exact solution of the nonlinear governing equations including Coriolis and centripetal terms for internal azimuthal waves with a uniform current in the $\beta$-plane approximation near the equator. This solution is described in the Lagrangian framework. The unidirectional azimuthal internal trapped are symmetric about the equator and propagate eastward above the thermocline and beneath the near-surface layer.
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