Abstract We study reggeon quantum mechanics when four-reggeon couplings are present as well as the conventional three-reggeon coupling. We show that as the reggeon intercept is increased an infinite number of states pass through energy zero. We give the values of the intercepts at which states pass through zero, construct the states explicitly, and tell how fast their energy changes with intercept. We show that these higher states do not interfere with the construction of spin models. However, no spin model is possible if the one-reggeon ↔ three-reggeon coupling is non-zero.