On the Bäcklund transformations of the Riccati equation: the differential-geometric approach revisited (Q2577542)

The author asks the following simple question: given two functions \(a, b\in C^{1}(\mathbb{R},\mathbb{R}),\) and \(n\geq 2,\) when can we solve by quadrature the generalized Riccati-Abel equation \(y^{\prime }=y^{n}+a(x)y+b(x)?\) For the simple case \(n=2\) and \(a=0,\) the problem goes back to Liouville, where methods of differential algebra were first used and later inverse spectral transform. For \(n=2,\;3\) one needs to use methods of symplectic geometry, the so-called Novikov-Bogoyavlensky-Moser approach for the reduction of nonlinear dynamical systems. In this paper, the author revisits the problem with new tools such as the Bäcklund transformation and new differential-geometric properties of Lax-type operators.