  In Part I of our study on 2D Euler equation, we established the spectral
theorem for a linearized 2D Euler equation. We also computed the point spectrum
through continued fractions, and identified the eigenvalues with nonzero real
parts.
  In this Part II of our study, first we discuss the Lax pairs for both 2D and
3D Euler equations. The existence of Lax pairs suggests that the hyperbolic
foliations of 2D and 3D Euler equations may be degenerate, i.e., there exist
homoclinic structures. Then we investigate the question on the degeneracy v.s.
nondegeneracy of the hyperbolic foliations for Galerkin truncations of 2D Euler
equation. In particular, for a Galerkin truncation, we have computed the
explicit representation of the hyperbolic foliation which is of the degenerate
case, i.e., figure-eight case. We also study the robustness of this degeneracy
for a so-called dashed-line model through higher order Melnikov functions. The
first order and second order Melnikov functions are all identically zero, which
indicates that the degeneracy is relatively robust. The study in this paper
serves a clue in searching for homoclinic structures for 2D Euler equation. The
recent breakthrough result of mine on the existence of a Lax pair for 2D Euler
equation, strongly supports the possible existence of homoclinic structures for
2D Euler equation.
