  We study the transfer matrix spectral problem for the cyclic representations
of the trigonometric 6-vertex reflection algebra associated to the
Bazhanov-Stroganov Lax operator. The results apply as well to the spectral
analysis of the lattice sine-Gordon model with integrable open boundary
conditions. This spectral analysis is developed by implementing the method of
separation of variables (SoV). The transfer matrix spectrum (both eigenvalues
and eigenstates) is completely characterized in terms of the set of solutions
to a discrete system of polynomial equations in a given class of functions.
Moreover, we prove an equivalent characterization as the set of solutions to a
Baxter's like T-Q functional equation and rewrite the transfer matrix
eigenstates in an algebraic Bethe ansatz form. In order to explain our method
in a simple case, the present paper is restricted to representations containing
one constraint on the boundary parameters and on the parameters of the
Bazhanov-Stroganov Lax operator. In a next article, some more technical tools
(like Baxter's gauge transformations) will be introduced to extend our approach
to general integrable boundary conditions.
