The equilibrium structural parameters, symmetry groups, and dipole moments of the free hollow electron-deficient TlBi ₁₉, Tl ₂Bi ₁₈, p-, m-, o-Tl ₂Pb ₁₀ shells and the endohedral (He, Ne)@Tl ₂Pb ₁₀, (Y, La, Lu, Tl, Bi, Ce)@TlPb ₁₁, Tl@(Y, La, Lu) Pb ₁₁, Sr@TlBi ₁₉, Tl@SrBi ₁₉, Yb@TlBi ₁₉, Tl@YbBi ₁₉, [Y@TlBi ₁₉] ⁺, [Lu@TlBi ₁₉] ⁺, Sr@Tl ₂Bi ₁₈, and Yb@Tl ₂Bi ₁₈ complexes are predicted by using DFT (U)PBE0 hybrid method. The isomerization of (Sr, Yb)@Tl _kBi _20–k into Tl@(Sr, Yb) Tl _k–1Bi _20–k is endothermic. The isodesmic substitution of an endo-atom in the Tl@TlPb ₁₁ cluster for a rare-earth atom is exothermic. The hollow Tl _kBi _20–k (Sr, Yb) complexes with Tl in the shell and with Sr or Yb exo-atom are impossible. The endo-atoms plunge in hollow TlPb ₁₁ and hollow TlBi ₁₉ shells without delay on its surface. The (C _5v)-TlPb ₁₁ and (C _3v)-TlBi ₁₉ shells are stabilized by an endo-atom X with oxidation number Ξ _X = +3 and +2, respectively. No one cerium IV oxidation state has been predicted. The integer number Ξ _X is presented as a function of the populations of one-electron states in the shell surrounding the endo-atom X. Each Pb or Bi atom in the electron-deficient clusters retains an entirely localized lone electron pair. The change in the IR spectrum when a metal atom is embedded into its cavity indicates a strong interaction between the components of the endohedral complex. Structural aspects of the β-decay of a radioactive endo-atom in the cavity of a metal cluster are discussed.