Unique asymptotics of $SO(k)\times SO(n-k+1)$ symmetric ancient ovals of Ricci flow

We obtain the unique asymptotics of $SO(k)\times SO(n-k+1)$-invariant, compact, simply-connected, factorwisely non-self-similar $n$-dimensional $\kappa$-solutions of the Ricci flow $(M^n,g(t))$, where $n\ge 4$ and $2\le k\le n-2$. More precisely, these $\kappa$-solutions are either ancient ovals of the Ricci flow that are diffeomorphic to the standard sphere $S^n$, having a positive curvature operator metric $g(t)$ and a cylindrical tangent flow at $-\infty$, or they are a Riemannian product of a Perelman's ancient oval and a shrinking round sphere. The metric $g(t)$ of every $SO(k)\times SO(n-k+1)$-invariant ancient oval is represented in the form $g(t)=dz\otimes dz+F^2(z,t)g_{S^{k-1}}+G^2(z,t)g_{S^{n-k}}$ (up to flipping $k-1$ and $n-k$). We obtain results about the blowdown limits of such solutions, establish the unique sharp asymptotics of the profile function $G(z,t)$, and prove that the uniqueness of $G(z,t)$ implies the uniqueness of $F(z,t)$. In particular, this provides the first instance of a classification result for geometric flows represented by a coupled PDE system, opening new avenues for studying the classification of higher-dimensional $\kappa$-solutions of the Ricci flow.