Dynamical systems often have structural features that incorporate underlying physics and conservation laws that reflect basic properties of phenomena of interest. Reduced models for these dynamical systems that do not share such key structural features, even if they otherwise have high fidelity, may produce responses that are ``unphysical" and as a result may be unsuitable for use as dependable surrogates. We seek systems that have structure characterized as either port-Hamiltonian or second-order (or both), and that, within the latitude allowed by those constraints, is also a best possible approximation to the original system as discerned by the ℋ₂ error measure. In this work, we develop necessary optimality conditions that must be satisfied by such reduced systems.