J Austral Math Soc Ser B 34 pp--, 1992.
(Received 28 June 1990; revised 15 January 1991)
In the present paper, we make use of the method of asymptotic integration to get estimates on those regions in the complex plane where singularities and critical points of solutions of the Matrix-Riccati differential equation with polynomial coefficients may appear. The result is that most of these points lie around a finite number of permanent critical directions. These permanent directions are defined by the coefficients of the differential equation. The number of singularities outside certain domains around the permanent critical directions, in a circle of radius r, is of growth O(log r). Applications of the results to periodic solutions and to the determination of critical points are given.