A NOTE ON QUASI-PERIODIC SOLUTIONS OF SOME ELLIPTIC-SYSTEMS

We extend a recent method of proof of a theorem by Kolmogorov on the c onservation of quasi-periodic motion in Hamiltonian systems so as to p rove existence of (uncountably many) real-analytic quasi-periodic solu tions for elliptic systems Delta u = epsilon f(x)(u, y), where u: y is an element of R(M) --> u(y) is an element of R(N), f = f(x, y) is a r eal-analytic periodic Function and epsilon is a small parameter. Kolmo gorov's theorem is obtained (in a special case) when M = 1 while the c ase N = 1 is (a special case of) a theorem by J. Moser on minimal foli ations of codimension 1 on a torus T-M + 1 In the autonomous case, f = f(x), the above result holds for any epsilon.