Our paper addresses characterizing conditions for linear quadratic (LQ) games to be potential. The desired properties of potential games in finite action settings, such as convergence of learning dynamics to Nash equilibria and challenges of learning Nash equilibria in continuous state and action settings, motivate us to characterize LQ potential games. Our first contribution is to consider two-player LQ games with full-state feedback and scalar states and action space, and to analytically verify that the set of potential games within this example is limited, essentially differing only slightly from an identical interest game. Given this finding, we restrict the class of LQ games to those with decoupled dynamics and decoupled state linear feedback information structure. For this subclass, we show that the set of potential games strictly includes non-identical interest games and characterize conditions for LQ games in this subclass to be potential. We further derive their corresponding potential function and prove the existence of a Nash equilibrium. Meanwhile, we highlight the challenges in characterizations of the Nash equilibrium for this class of potential LQ games, theoretically and through simulation. (c) 2025 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).