The homology-invariant equation can be regarded as the autonomous pair of equations d U d log ⁡ ⁡ ξ ξ = − − U ( U + n ( n + 1 ) − − 1 V − − 3 ) and d V d log ⁡ ⁡ ξ ξ = V ( U + ( n + 1 ) − − 1 V − − 1 ) .