On integrable by quadratures generalized Riccati-Abel equations: Differential-geometric and Lie-algebraic analysis

More than one hundred and fifty years ago J. Liouville posed the problem of describing Riccati equations dydx = y2 + a (x) y + b (x) which are integrable by quadratures. But up to now there exists no effective theory answering the question whether a given Riccati equation is integrable or not. Based on the theory of Lax type integrable dynamical systems, eighteen years ago a new attempt was made to study the Liouville problem. A new approach was devised to investigate the integrability by quadratures by reducing a given Riccati equation dydx = y2 + f (x) to some equivalent nonlinear evolution equations in partial derivatives with Cauchy-Goursat initial data, and proving further their Lax type integrability, connected via Liouville with the integrability by quadratures [6, 8]. This approach having background in modern differential-geometric and Lie-algebraic techniques, was developed before by F. Estabrook, H. Wahlquist, S. Novikov, V. Marchenko for the well-known Korteveg-de Vries type equations. In this report we apply these methods to study integrability by quadratures of a generalized Riccati-Abel equation.