Bifurcations of planar balanced configurations for the $n$-body problem in $\mathbb{R}^4$

Central configurations play a fundamental role in the Newtonian $n$-body problem, as they give rise to motions in which the configuration evolves while preserving its shape up to rotation and scaling. These include relative equilibria, where the configuration rigidly rotates about the center of mass and each body moves along a circular orbit. For $d\le3$, such motions originate only from planar central configurations, whereas in higher dimensions the richer structure of the orthogonal group admits new balanced configurations that can produce non-planar relative equilibria. Building on the framework introduced by Asselle, Portaluri and Fenucci [J. Fixed Point Theory App., 2022], we analyze bifurcations of planar balanced configurations in $\mathbb{R}^4$. We extend a classical variational result, which guarantees the existence of bifurcation points along trivial branches of critical points that are degenerate only at finitely many points, to the case where the trivial branch remains degenerate throughout. Applying this extension, we establish the existence of bifurcation points along the planar balanced configuration branch and derive a lower bound on their number.