Rigid systems of second-order linear differential equations

We say that a system of differential equationsx¨(t)=Ax˙(t)+Bx(t)+Cu(t),A,B∈Cm×m,C∈Cm×n,is rigid if it can be reduced by substitutionsx(t)=Sy(t),u(t)=Uy˙(t)+Vy(t)+Pv(t),with nonsingular S and P to each system obtained from it by a small enough perturbation of its matrices A, B, C. We prove that there exists a rigid system for given m and n   if and only if m