  Based on an accurate determination of the potential energy surfaces of
Rb$_3^+$ correlated to its first asymptotic limit Rb$^+$$+$Rb($5s$)$+$Rb($5s$),
we identify the presence of intersections of a pair of singlet and triplet
surfaces over all interparticle distances, leading to Jahn-Teller couplings. We
elaborate scenarios for charge exchange between ultracold charged atom-dimer
complex (Rb$+$Rb$_2^+$ or Rb$^+$$+$Rb$_2$), predicting a strong selectivity on
the preparation of the initial state of the dimer. We also demonstrate that the
JT couplings must drive the three-body recombination (TBR) of Rb$^+$, Rb, and
Rb at ultracold energies. Using the current analysis, we provide a consistent
picture of the TBR experiments performed in ion-atom hybrid Rb samples
\cite{dieterle2020inelastic,harter2012single}. We also demonstrate the presence
of JT coupling as a general phenomenon in the singly-charged homonuclear alkali
triatomic systems.
