  A phenomenon of emergence of stability islands in phase-space is reported for
two periodic potentials with tiling symmetries, one square and the other
hexagonal, inspired by bidimensional Hamiltonian models of optical lattices.
The structures found, here termed as island myriads, resemble web-tori with
notable fractality and arise at energy levels reaching that of unstable
equilibria. In general, the myriad is an arrangement of concentric island
chains with properties relying on the translational and rotational symmetries
of the potential functions. In the square system, orbits within the myriad come
in isochronous pairs and can have different periodic closure, either returning
to their initial position or jumping to identical sites in neighbor cells of
the lattice, therefore impacting transport properties. As seen when compared to
a more generic case, i.e. the rectangular lattice, the breaking of square
symmetry disrupts the myriad even for small deviations from its equilateral
configuration. For the hexagonal case, the myriad was found but in attenuated
form, mostly due to extra instabilities in the potential surface that prevent
the stabilization of orbits forming the chains.
