In this paper, we study equi-centro-affine extremal hypersurfaces in an ellipsoid. By investigating the evolution of invariant submanifold flows under centro-affine unimodular transformations, we derive the first and second variational formulas for the associated invariant area. Stability analysis reveals that the circles with radius \(r=\sqrt{6}/3\) on \({\mathbb {S}}^2(1)\) are characterized as being equi-centro-affine maximal. Moreover, we provide a classification of the compact isoparametric equi-centro-affine extremal hypersurfaces on \((n+1)\) -dimensional sphere, as well as the generalized closed equi-centro-affine extremal curves on 2-dimensional sphere. These curves are shown to belong to a family of transcendental curves \(\textrm{x}_{p,q}\) (p and q are two coprime positive integers satisfying that \(1/2<p/q<1\) ). In addition, we establish an equi-centro-affine version of isoperimetric inequality on \({\mathbb {S}}^2(1)\) under certain assumptions.
