  The nonrelativistic energies of the homonuclear ion T$_2^+$ are calculated
for the ground state using the Lagrange-mesh method as was done for the
isotopomers H$_2^+$ and D$_2^+$ ({\it J. Phys. B: At. Mol. Opt. Phys.} {\bf 45}
065101 and {\it J. Phys. B: At. Mol. Opt. Phys.} {\bf 46} 245101). Energies and
eigenfunctions are obtained up to four of the lowest bound vibrational states
($v=0,1,2,3$) which support 62, 61, 60 and 58 bound rotational states,
respectively. Some quasibound sates are also presented until $L =$ 68. From the
obtained wave functions, electric quadrupole transitions per time unit are
calculated between those states over the whole rotational bands. Extensive
results are presented with 6 significant digits. The ground state energy of the
symmetric systems $^{\infty}$H$_2^+$, $^{\infty}$H$^-$, Ps$^{-}$, H$^-$, D$^-$,
T$^-$, $\mu^+\mu^+e$ and $\mu^+e\,e$ is presented with high accuracy as a
function of $\beta=m_3/(2m+m_3)$.
