  Eigenstates of the FQHE hamiltonian problem after to be projected on the LLL
are determined for filling factors 1/q, with q an odd number. The solutions are
found for an infinite class of finite samples in which the Coulomb potential is
periodically extended. Therefore, a thermodynamic limit solution is also
identified. The results suggest the presence of integrability properties in
FQHE systems. The many particle states are simple Slater determinants
constructed with special single particle states. These orbitals are defined as
powers of order q of "composite fermion" like wavefunctions associated to a
reduced magnetic field B/q. At the same time, those "composite fermion" states
were obtained by factorizing and canceling fixed position (quasi-momentum
independent) zeros in previously derived exact Hartree-Fock orbitals. A formula
for the energy per particle of the FQHE states is given for finite samples as
well as for the thermodynamic limit state. As a side result, the same
"composite fermions" like orbitals are employed to construct variational
wavefunctions of the system, showing zeros of order q as two electrons approach
each other, as Laughlin states do. The long range spatial correlation
associated to the starting HF solutions may further reduce the energy of these
states.
