  Dynamical symmetries of Hamiltonians quantized models of discrete non-linear
Schroedinger chain (DNLS) and of Ablowitz-Ladik chain (AL) are studied. It is
shown that for $n$-sites the dynamical algebra of DNLS Hamilton operator is
given by the $su(n)$ algebra, while the respective symmetry for the AL case is
the quantum algebra su_q(n). The q-deformation of the dynamical symmetry in the
AL model is due to the non-canonical oscillator-like structure of the raising
and lowering operators at each site.
  Invariants of motions are found in terms of Casimir central elements of su(n)
and su_q(n) algebra generators, for the DNLS and QAL cases respectively.
Utilizing the representation theory of the symmetry algebras we specialize to
the $n=2$ quantum dimer case and formulate the eigenvalue problem of each dimer
as a non-linear (q)-spin model. Analytic investigations of the ensuing
three-term non-linear recurrence relations are carried out and the respective
orthonormal and complete eigenvector bases are determined.
  The quantum manifestation of the classical self-trapping in the QDNLS-dimer
and its absence in the QAL-dimer, is analysed by studying the asymptotic
attraction and repulsion respectively, of the energy levels versus the strength
of non-linearity. Our treatment predicts for the QDNLS-dimer, a
phase-transition like behaviour in the rate of change of the logarithm of
eigenenergy differences, for values of the non-linearity parameter near the
classical bifurcation point.
