  This study investigates the properties of the $T_{\psi\psi}$ tetraquark
candidates within a coupled-channels calculation of the $c\bar c- c\bar c$
system, specifically focusing on the $J^P=0^\pm$, $1^\pm$, and $2^\pm$ sectors.
The analysis includes various channels containing a $J/\psi$, $\psi^\prime$,
$\eta_c$, and $\eta_c^\prime$ meson. By searching for poles in the scattering
matrix, a total of 29 states in different $J^P$ sectors with masses ranging
from 6.1 to 7.6 GeV/c$^2$ are identified. The study further investigates the
masses, widths and branching ratios of these states, leading to the
identification of two potential candidates for the experimental
$T_{\psi\psi}(6200)$ tetraquark, one candidate for $T_{\psi\psi}(6600)$, two
for $T_{\psi\psi}(6700)$, four for $T_{\psi\psi}(6900)$, and three for
$T_{\psi\psi}(7200)$ tetraquarks. Additionally, the paper discusses strategies
to discriminate between different candidates and explores possible detection
channels for further $c\bar c- c\bar c$ states.
