I classify all smooth non‑constant solutions of the second‑order wave equation \ (ₔₕ=H () \) of the form \ ( (u, v) =f ( (u, v) ) \) that constitute a genuine ODE reduction. The definition is formulated through the existence of an auxiliary variable \ ( (u, v) \) such that the coefficients \ (ᵤᵥ\) and \ (ₔₕ\) depend only on \ (\), with \ (ᵤᵥ0\), and \ (\) is a strictly monotone function of \ (\). From this definition we prove rigorously that the geometric coefficients \ (P=ᵤᵥ\) and \ (Q=ₔₕ\) must themselves be functions of \ (\) alone --- the constrained condition. Under that condition a complete integration shows that \ (\) is necessarily a strictly monotone function of either a linear expression \ (= u+ v+\) (with \ (0\) ) or a factorizable bilinear expression \ (= (u+) (v+) \) (with \ (0\) ). The reduced ODE for \ (g () =f ( () ) \) is the generalized oscillator equation \ (g'' = H (g) \) in the linear case and the generalized Amsler equation \ ( (g') ' = c\, H (g) \) in the bilinear case. The so‑called liberated form \ (ₔₕ+k () ᵤᵥ = L () \) is shown to be a universal reparametrisation that alone does not guarantee a reduction; an explicit counterexample is given. The classification is complete for smooth data; the extension to complex holomorphic functions is discussed as a remark. The proof is self‑contained and uses only smoothness and elementary calculus, providing a direct alternative to the classical Lie symmetry approach. We additionally prove that the only solution that can be simultaneously represented as a function of a linear phase and of a bilinear phase on overlapping open sets is the constant vacuum \ (₀\) with \ (H (₀) =0\). This result clarifies the role of the constant solution as the unique possible bridge between the two reduction families.
Anton Kalmykov (Mon,) studied this question.