  In the model suggested by Smilansky one studies an operator describing the
interaction between a quantum graph and a system of $K$ one-dimensional
oscillators attached at several different points in the graph. The present
paper is the first one in which the case $K>1$ is investigated. For the sake of
simplicity we consider K=2, but our argument is of a general character. In this
first of two papers on the problem, we describe the absolutely continuous
spectrum. Our approach is based upon scattering theory.
