Allowed Quantum States- Filkelstein and Rubinstein constraints

So basically i'm doing a report on Finkelstein and Rubinstein constraints. I have a system where the allowed quantum states satisfy $e^{i\frac{2\pi}{3\sqrt{3}}(L_{1}+L_{2}+L_{3})}e^{i\frac{2\pi}{3\sqrt{3}}(K_{1}+K_{2}+K_{3})}|\psi\rangle$ = $+1|\psi\rangle$ and $e^{−iπL_{3}}e^{−iπK_{3}} |ψ⟩ = +1|ψ⟩$. The allowed states are of the form $|J,L_{3}\rangle\otimes|I,K_{3}\rangle$. My question is how do i find these allowed states? The paper that i've read says I just need to find the simultaneous eigenvalues that satisfy both constraint but how do I find these eigenvalues?

-
Sorry, what are the $L_i$ and $K_i$ here? –  Michael Brown Mar 5 at 1:51