On invariant tori in some reversible systems

In the present paper, we consider the reversible system $ẋ=ω0+f(x,y)$⁠, $ẏ=g(x,y)$⁠, where x ∈ T^d, y ∽ 0 ∈ R^d, ω₀ is Diophantine, f(x, y) = O(y), g(x, y) = O(y²) and f, g are reversible with respect to the involution G: (x, y) ↦ (−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 Γ₀ of the reversible system with Diophantine frequency ω₀ by other invariant tori. We will prove that if the Birkhoff normal form around Γ₀ is 0-degenerate, then Γ₀ 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 Γ₀ being one. We will also prove that if the Birkhoff normal form around Γ₀ is j-degenerate (1 ≤ j ≤ d − 1) and condition (1.6) is satisfied, then through Γ₀ there passes an analytic subvariety of dimension d + j foliated into analytic invariant tori with frequency vector ω₀. If the Birkhoff normal form around Γ₀ is d − 1-degenerate, we will prove a stronger result, that is, a full neighborhood of Γ₀ is foliated into analytic invariant tori with frequency vectors proportional to ω₀.