Trojan Wave Packets in the Circularly Polarized and the Magnetic Fields on the multi-layer Langmuir type $(1)$ Helium trajectories with the inter-layer polygon type change

Some time ago we have discovered the theorem stating that the number of a distinct classically rotating frozen electron configurations in the external magnetic field, the nuclear ion field and the co-rotating electric field may be at least the product of all foldings of total force surfaces (the maximal number any straight line may cut the surface which notrivially is 2) obtained by putting each of the multi-dimensional gradient component of the Zero Velocity Surface to zero. Consistently with it we have discovered a several highly symmetric configurations originated from the Langmuir type $(1)$ ``Hoop Earrings" rotating Helium-like model trajectories [1] with further small discrete rotational symmetry distortion by the Circularly Polarized electromagnetic field without the loss of its nature. Those were the multi-electron atom trajectories consisting of two or many layers of electrons moving in phase on circles with electron configurations placed at the vertexes of angles of identical by type but only scaled regular polygons parallel in space which additionally could be also symmetrically twisted between each other. Here we extend the case to the situation when the neighboring layer polygons are plane-symmetrically of a different kind. For example the hexagon in the central plane may accompany two triangles placed symmetrically one in the upper paraller plane and the other in the lower or the octagon may similarly neighbor two squares etc. The classical stabilization of the trajectories by the combination of fields further leads to the existence of non-dispersing localized Trojan-like wave packets moving around the trajectories. The time dependent Hartree simulations of such Wave Packets as well as those with the Time Dependent Quantum Diffusion Monte Carlo Method are conducted. [1] M. Kalinski, et al., Phys. Rev. Lett. {bf 95}, 103001, (2005).