Double pendulum is non-ergodic?

Question

I got really curious about ergodicity in double pendulum. The diagram below represents the time it takes for a double pendulum to "flip" when started from different initial positions. The white color on the graph corresponds to the positions where the pendulum doesn't flip during the simulation time (of about 2000 swings). While the central white region of the graph is intuitive and corresponds to states in which the pendulum does not have enough energy to flip (the region corresponds to inequality $3 \cos \theta_1+\cos\theta_2>2$), there are non-flipping states outside the region. These states do not flip, even though they have enough energy, which indicates the behavior of the system starting from such states is possibly non-ergodic. The existence of unreachable regions in the phase space would seem to imply the existence of non-trivial integrals of motion (for instance, an indicator function that is equal to 1 in the unreachable region of the phase space, and 0 everywhere else). My question is what these non-trivial integrals of motion could possibly be like? Can we find them somehow? Or will the pendulum eventually flip everywhere where it can flip, but just in an extremely long time?