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In this paper we construct small amplitude periodic internal waves travelingat the boundary region between two rotational and homogeneous fluids withdifferent densities. Within a period, the waves we obtain have the propertythat the gradient of the stream function associated to the fluid beneath theinterface vanishes, on the wave surface, at exactly two points. Furthermore,there exists a critical layer which is bounded from above by the wave profile.Besides, we prove, without excluding the presence of stagnation points, that ifthe vorticity function associated to each fluid in part is real-analytic,bounded, and non-increasing, then capillary-gravity steady internal waves are apriori real-analytic. Our new method provides the real-analyticity of capillaryand capillary-gravity waves with stagnation points traveling over a homogeneousrotational fluid under the same restrictions on the vorticity function.

Steady internal water waves with a critical layer bounded by the wave surface