On invariant tori in some reversible systems

In the present paper, we consider the reversible system x=omega 0+f(x,y), y=g(x,y), where x is an element of T-d, y similar to 0 is an element of R-d, omega(0) is Diophantine, f(x, y) = O(y), g(x, y) = O(y(2)) and f, g are reversible with respect to the involution G: (x, y) bar right arrow (-x, y), that is, f(-x, y) = f(x, y), g(-x, y) = -g(x, y). We study the accumulation of an analytic invariant torus Gamma(0) of the reversible system with Diophantine frequency omega(0) by other invariant tori. We will prove that if the Birkhoff normal form around Gamma(0) is 0-degenerate, then Gamma(0) is accumulated by other analytic invariant tori, the Lebesgue measure of the union of these tori being positive and the density of the union of these tori at Gamma(0) being one. We will also prove that if the Birkhoff normal form around Gamma(0) is j-degenerate (1 <= j <= d - 1) and condition (1.6) is satisfied, then through Gamma(0) there passes an analytic subvariety of dimension d + j foliated into analytic invariant tori with frequency vector omega 0. If the Birkhoff normal form around Gamma(0) is d - 1-degenerate, we will prove a stronger result, that is, a full neighborhood of Gamma(0) is foliated into analytic invariant tori with frequency vectors proportional to omega(0).