Solvability and integrability of reduced projective‑invariant ODEs obtained from invariants

Determine whether the reduced ordinary differential equations obtained from the projective‑invariant ODE classes—specifically, the second‑order ODE in the variable V(S) that results from the sixth‑order equation defined by the relation between the invariants 22 = F(21) under the substitutions S(x) = u_{xxx}/u_x − (3/2)(u_{xx}/u_x)^2, W(S) = S_x/S^{3/2}, and V(S) = W(S)^2, and the third‑order ODE in W(S) that results from the seventh‑order equation defined by 23 = F(21,22)—are exactly solvable or integrable, and, if so, identify for which choices of the function F this holds.

Background

In the setting of a projective (Möbius) transformation acting on x and u(x), the authors compute three invariants expressible via the Schwarzian derivative S[u]. Using these, they construct invariant ODEs of orders five, six, and seven. The fifth‑order case is shown to be solvable in general.

For the higher‑order cases, they reduce the sixth‑order equation (arising from 22 = F(21)) to a second‑order ODE in V(S) = (S_x/S^{3/2})^2, and the seventh‑order equation (arising from 23 = F(21,22)) to a third‑order ODE in W(S) = S_x/S^{3/2}. Whether these reduced equations are exactly solvable or integrable remains unresolved.