Dispersion law for a tight binding Hamiltonian and particle states at $t$$rightarrow$$infty$

A spinless fermion (possessing an electric charge) can move across the sites of
the discrete (translationally-invariant) lattice. The structure
features three kinds of sites: $α_n$, $β_n$, $γ_n$ with $n = 1,…,N$ and $N ≫ 1$.
For each n, $α_n$, $β_n$ and $γ_n$ share the same x-coordinate $t_n$ = $na$. The particle can directly hop between nearest-neighbour sites $α_n$ and $α_n$$_+$$_1$, between sites $α_n-β_n$ (same n) and between sites $α_n-γ_n$ (same n).

Due tight-binding model I derived the energy spectrum of the particle, showing that it comprises three energy bands $E_+(k)$, $E_-(k)$ and $E_0(k)=0$. Now, I have to find the dispersion laws.
Why do I have a flat energy band? What is its physical significance?. Also, what happens to the particle to $t rightarrow infty$ if the state at $t = 0$ is in a generic site $|Ψ(0)> = |β_m>$?