  The Davydov model describes amide I energy transfer in proteins without
dispersion or dissipation. In spite of five decades of study, there are few
exact analytical results, especially for the discrete version of this model.
Here we develop two methods to determine the exact orthonormal, multi-quanta,
eigenstates of the Davydov dimer. The first method involves the integration of
a system of ordinary differential equations and the second method applies
purely algebraic methods to this problem. We obtain the general expression of
the eigenvalues for any number of quanta and also, as examples, apply the
methods to the detailed derivation of the eigenvectors for one to four quanta,
plus a brief example in the case of $n=5$ and $n=6$.
