The dynamics of the Ehrhard–Müller system with invariant algebraic surfaces

The paper’s Theorem 1 classifies all quadratic invariant algebraic surfaces of the Ehrhard–Müller system and lists exactly the four families (a)–(d) with cofactors k ∈ {−1, −2}, matching the candidate’s solution one-for-one (Table 1: x^2−z for s=1/2; y^2+z^2−cx for s=2,r=0; y^2+z^2 for r=c=0; y^2+z^2−r x^2 for s=1,c=0) . The paper’s proof substitutes a general quadratic f and an affine cofactor k into Xf=kf, derives a 20-equation linear system, shows the cofactor’s linear part vanishes (k1=k2=k3=0), and then resolves the remaining cases to obtain (a)–(d) . The candidate independently reaches the same list by (i) forcing the cofactor to be constant via homogeneous-cubic matching, (ii) reducing f2 to Ax^2+B(y^2+z^2), and (iii) solving the coefficient system, with a clear sufficiency check. One minor gap in the paper is that the statement purports to cover all real r,s,c, yet the degenerate case s=0 implies ẋ=0, so any quadratic in x alone is a (k≡0) Darboux polynomial; this family is omitted in Table 1, whereas the candidate flags it explicitly as a trivial/degenerate exception. This can be fixed by stating s≠0 or excluding k≡0 first-integrals from the classification. Aside from this edge case, the proofs agree and the main theorem matches the candidate’s result .