In 1982, Wu-Yi Hsiang, in an article published in the Journal of Differential Geometry, classified constant mean curvature hypersurfaces in Euclidean space, invariant by the action of the group \(O(p)\times O(q).\) In his work, he conjectured that there is only one of such hypersurfaces in the Euclidean space, invariant by the action of the group \(O(p)\times O(q),\) whose profile curve has a singularity at the origin. In this paper, we prove this conjecture using blowing up techniques for degenerate singularities and invariant manifold theory for a tridimensional system of ordinary differential equations. We remark that the noninvariance of the constant mean curvature equation by homotheties prevents us from using the method developed by E. Bombieri, E. De Giorgi, and E. Giusti, in an article published in the Inventiones Mathematicae in 1969, to classify minimal hypersurfaces in the Euclidean space and transform the constant mean curvature equation in a bidimensional system of ordinary differential equations. This forces us to analyze a tridimensional system of ordinary differential equations with degenerate singularities.
