  A discrete nonlinear system is analysed in case of open chain boundary
conditions at the ends. It is shown that the integrability of the system
remains intact, by obtaining a modified set of Lax equations which
automatically take care of the boundary conditions. The same Lax pair also
conforms to the conditions stipulated by Sklyanin [5]. The quantum inverse
problem is set up and the diagonalisation is carried out by the method of
sparation of variables. B\"{a}cklund transformations are then derived under the
modified boundary conditions using the classical r-matrix . Finally by
quantising the B\"{a}cklund transformation it is possible to identify the
relation satisfied by the eigenvalue of Baxter's Q-operator even for the quasi
periodic situation.
