  In this paper we try to find examples of integrable natural Hamiltonian
systems on the sphere $S^2$ with the symmetries of each Platonic polyhedra.
Although some of these systems are known, their expression is extremely
complicated; we try here to find the simplest possible expressions for this
kind of dynamical systems. Even in the simplest cases it is not easy to prove
their integrability by direct computation of the first integrals, therefore, we
make use of numerical methods to provide evidences of integrability; namely, by
analyzing their Poincar\'e sections (surface sections). In this way we find
three systems with platonic symmetries, one for each class of equivalent
Platonic polyhedra: tetrahedral, exahedral-octahedral,
dodecahedral-icosahedral, showing evidences of integrability. The proof of
integrability and the construction of the first integrals are left for further
works. As an outline of the possible developments if the integrability of these
systems will be proved, we show how to build from them new integrable systems
in dimension three and, from these, superintegrable systems in dimension four
corresponding to superintegrable interactions among four points on a line, in
analogy with the systems with dihedral symmetry treated in a previous article.
A common feature of these possibly integrable systems is, besides to the rich
symmetry group on the configuration manifold, the partition of the latter into
dynamically separated regions showing a simple structure of the potential in
their interior. This observation allows to conjecture integrability for a class
of Hamiltonian systems in the Euclidean spaces.
