New branch of Kaluza-Klein compactification

We found a new branch of solutions in Freund-Rubin type flux compactifications. The geometry of these solutions is described as the external space which has a de Sitter symmetry and the internal space which is topologically spherical. However, it is not a simple form of ${\mathrm{dS}}_p\times S^q$ but a warped product of de Sitter space and a deformed sphere. We explicitly constructed numerical solutions for a specific case with $p=4$ and $q=4$. We show that the new branch of solutions emanates from the marginally stable solution in the branch of ${\mathrm{dS}}_4\times S^4$ solutions.