  Using the Wei-Norman theory we obtain a time-dependent complex Riccati
equation (TDCRE) as the solution of the time evolution operator (TEO) of
quantum systems described by time-dependent (TD) Hamiltonians that are linear
combinations of the generators of the $\mathfrak{su}(1,1)$, $\mathfrak{su}(2)$
and $\mathfrak{so}(2,1)$ Lie algebras. Using a recently developed solution for
the time evolution of these quantum systems we solve the TDCRE recursively as
generalized continued fractions, which are optimal for numerical
implementations, and establish the necessary and sufficient conditions for the
unitarity of the TEO in the factorized representation. The inherited symmetries
of quantum systems can be recognized by a simple inspection of the TDCRE,
allowing effective quantum Hamiltonians to be associated with it, as we show
for the Bloch-Riccati equation whose Hamiltonian corresponds to that of a
generic TD system of the Lie algebra $\mathfrak{su}(2)$. As an application, but
also as a consistency test, we compare our solution with the analytic one for
the Bloch-Riccati equation considering the Rabi frequency driven by a complex
hyperbolic secant pulse generating spin inversion, showing an excellent
agreement.
