It is known that there is a one-to-one mapping between oriented directed graphs and zero-sum replicator dynamics (Lotka-Volterra equations). We investigate the problem of determining whether these dynamics are Liouville-Arnold integrable. Building on prior work, we establish two families of integrable graphs. With these results and numerical methods, we classify the dynamics generated by almost all oriented directed graphs on six vertices, except for three graphs that are integrable but not part of the taxonomy.