The \(n\)-dimensional homogeneous Lotka–Volterra (HLV) ladder equation possesses Lie symmetries that generate a Lie algebra isomorphic to \(\mathfrak{sl}_{n}(\mathbb{K})\) [J. Phys. Soc. Jpn. 71 (2002) 2396 and 72 (2003) 973], where \(\mathbb{K}=\mathbb{C}\) or \(\mathbb{R}\). In this work, using the Lie algebraic structure of the HLV ladder equation, we derive new \(n\)-dimensional dynamical systems and prove their integrability from the viewpoint of Lie symmetries.