On Exact and Approximate Symmetries of Algebraic and Ordinary Differential Equations with a Small Parameter

Both the paper and the model derive the same determining equation for the first-order BGI approximate symmetry in evolutionary form and conclude existence of ζ1 of order ≤ n−1. The paper argues existence by stating the PDE is in Cauchy–Kovalevskaya form with respect to x (Remark 6.1) leading to Theorem 6.1, while the model treats the equation as a linear ODE along the solution manifold’s total derivative D̄ with smooth coefficients, invoking standard ODE existence. Thus, they are consistent and correct, but use different existence arguments. See equations (6.9)–(6.11) and Theorem 6.1 in the paper , and the paper’s overview confirming the theorem’s scope for all orders n ≥ 2 .