Superposition formulas for rectangular matrix Riccati equations

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Abstract
A system of nonlinear ordinary differential equations allowing a superposition formula can be associated with every Lie group–subgroup pair G⊇G0. We consider the case when G=SL(n+k,C) and G0=P(k) is a maximal parabolic subgroup of G, leaving a k‐dimensional vector space invariant (1≤k≤n). The nonlinear ordinary differential equations (ODE’s) in this case are rectangular matrix Riccati equations for a matrix W(t)∈Cn×k. The special case n=rk (n,r,k∈N) is considered and a superposition formula is obtained, expressing the general solution in terms of r+3 particular solutions for r≥2, k≥2. For r=1 (square matrix Riccati equations) five solutions are needed, for r=n (projective Riccati equations) the required number is n+2.