The paper considers a singularly perturbed system with retarded argument \[ dx/dt=Ax_ t+By+E(x_ t,y,\epsilon),\quad dy/dt=Cx_ t+Dy+F(x_ t,y,\epsilon)\quad (0<\epsilon <<1) \] where \(x_ t=x(t+\theta)\) (-r\(\leq \theta \leq 0)\), A and C are linear bounded operators, B and D are matrices. The functions E and F are of \((2k+3)\) order and continuous differentiable for all arguments, moreover, \(| E(x,y,\epsilon)|\), \(| F(x,y,\epsilon)| \leq M\), \(\| E_ x(x,y,\epsilon)\|\), \(| E_ y(x,y,\epsilon)|\), \(\| F_ x(x,y,\epsilon)\|\), \(| F_ y(x,y,\epsilon)| \leq L\) for \((x,y,\epsilon)\in C^ n_{[-r,0]}\times R^ m\times [0,\epsilon_ 0]\). Here \(C^ n_{[- r,0]}\) is the space of continuous functions x: [-r,0]\(\to R^ n\) with norm \(\| x\| =\sup_{-r\leq \theta \leq 0}| x(\theta)|\). The paper proves an important conclusion: If there exist such M, L and \(\epsilon_ 1>0\) so that for \(0<\epsilon <\epsilon_ 1\) the system has exponential stable integral manifolds, and they may be represented by \[ x=f(u,\epsilon)=f^{(0)}(u)+\epsilon f^{(1)}(u)+...+\epsilon^{k- 1}f^{(k-1)}(u)+\epsilon^ kf^{(k)}(u,\epsilon) \]
\[ y=h(u,\epsilon)=h^{(0)}(u)+\epsilon h^{(1)}(u)+...+\epsilon^{k- 1}h^{(k-1)}(u)+\epsilon^ kh^{(k)}(u,\epsilon) \] where the functions \(f^{(i)}\) and \(h^{(i)}\) \((i=1,2,...,k-1)\) are continuous differentiable, \(f^{(k)}\) and \(h^{(k)}\) have continuous derivative on u for \(\epsilon\) fixed. Then the motion at manifolds is described by an ordinary differential equation \[ \dot u=Uu+\psi^ T(0)\{B[h(u,\epsilon)+D^{- 1}Cf(u,\epsilon)]+E[f(u,\epsilon),h(u,\epsilon),\epsilon]\} \] where the matrices U and \(\psi^ T(0)\) may be determined by the initial conditions of the system.