<p>In this paper, we study the multiplicity of solutions to the Yamabe problem on product manifolds. Specifically, we consider the product of a compact Riemannian manifold without boundary and with zero scalar curvature, and a compact Riemannian manifold with boundary, also with zero scalar curvature and constant mean curvature on the boundary. Our first objective is to demonstrate the existence of a sequence of bifurcation instants for a family of solutions to the Yamabe problem on the product manifold, using classical bifurcation theory. Our second objective is to classify a finite number of degeneracy instants that remained undecidable called neutral instants, using equivariant bifurcation techniques.</p>