Explicit solutions of the {\mathfrak{a}_1} -type lie-Scheffers system and a general Riccati equation

For a general differential system $ \dot{x}(t) = \sum\nolimits_{d = 1}^3 {u_d } (t){X_d} $ , where X _d generates the simple Lie algebra of type $ {\mathfrak{a}_1} $ , we compute the explicit solution in terms of iterated integrals of products of u _d ’s. As a byproduct we obtain the solution of a general Riccati equation by infinite quadratures.