The tight-binding model of electrons on a honeycomb lattice is studied in the presence of a homogeneous magnetic field. Provided the magnetic flux per unit hexagon is rational of the elementary flux the one-particle Hamiltonian is expressed in terms of the generators of the quantum group $U_q(sl_2)$. Employing the functional representation of $U_q(sl_2)$ the Harper equation is rewritten as a systems of two coupled functional equations on a complex plane. For the special values of quasi-momentum the entangled system admits solutions in terms of polynomials. In that case the system exhibits certain symmetry relations allowing to resolve the entanglement, and basic single equation determining the eigenvalues and eigenstates (polynomials) is obtained. Equations specifying the locations of the roots of polynomials on a complex plane and consequently the one-particle wave functions are found. Employing numeric analysis the roots of polynomials corresponding to different eigenstates are solved out and the diagrams exhibiting the ordered structure of one-particle states are depicted.