Symmetric Perfect $2$-colorings on $J(10,3)$

We study perfect $2$-coloring of the Johnson graphs $J(n,3)$ associated with the third largest eigenvalue and symmetric quotient matrix, which exists only when $n\in \{6,10\}$. We survey the known constructions in the case $n=6$, give a new construction for the two known perfect $2$-colorings in the case $n=10$, and prove that these are the only possible ones.