Demchenko’s nonholonomic case of gyroscopic ball rolling without sliding over a sphere after his 1923 Belgrade doctoral thesis

The paper accurately states Demchenko’s integrable case for a gyroscopic ball rolling without slipping on a sphere under the Zhukovsky condition C1 = A1 + A2, derives the core equations (constraints (21)–(22), dynamics (26)–(27)), and notes the elliptic reduction (dx/dt)^2 = X(x) (28) . The candidate’s method mirrors the modern Chaplygin-ball-with-rotor invariants F1–F3, which indeed hold for all ε , and correctly extracts the linear first integral J from the third equation in (27) . However, the candidate mis-specifies the contact-point angular momentum, effectively using G = A ω − D(ω,γ)γ and thus 2F2 = A(s^2+τ^2) + (A−D)n^2, instead of the correct spherical-inertia form G = (A+D)(s e_s+τ e_τ) + A n e_n, giving 2F2 = P(s^2+τ^2)+A n^2 with P=A+D (compare the general formula G = Iω − D(ω,γ)γ where here I = A Id + D E) . Consequently, the key cancellation step should use F3 − P·(2F2), not F3 − A·(2F2). Although the overall conclusion (elliptic integrability via a quartic X) still follows after correcting these coefficients, the candidate’s derivation as written relies on incorrect inertial factors.