Let $n+1 \geq 3$ be an integer and $p + q = n$. Inside the unit sphere $\mathbf{S}^{n+1}$ the product \begin{equation} \mathbf{S}^p(\sqrt{p/n}) \times \mathbf{S}^q(\sqrt{q/n}) = \{ (X,Y) \in \mathbf{R}^{p+1} \times \mathbf{R}^{q+1} \mid \lvert X \rvert^2 = p/n, \lvert Y \rvert^2 = q/n \}. \end{equation} defines an embedded minimal hypersurface.
They are known to be integrable, in the sense that every Jacobi field on $\Sigma_{p,q} = \mathbf{S}^q(\sqrt{p/n}) \times \mathbf{S}^q(\sqrt{q/n})$ is generated by a one-parameter family of minimal hypersurfaces.
Question. How is the integrability of $\Sigma_{p,q}$ proved? Is integrability known for any other embedded minimal hypersurface in $\mathbf{S}^{n+1}$?
The only reference I know for a proof of the integrability of $\Sigma_{p,q}$ is a paper of Allard and Almgren [1], which is however hard to read. (They also establish that product of three or more spheres - which however have higher codimension - are not integrable.)
[1] William Allard, Frederick Almgren Jr. On the Radial Behavior of Minimal Surfaces and the Uniqueness of their Tangent Cones. Annals of Mathematics, Second Series, Vol. 113, No. 2 (Mar., 1981), pp. 215-265.
