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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 outsidedoi:10.1017/s0334270000007402 fatcat:jxykqlkjhnbopjvrfgatdz3iue