  We define a general class of random systems of horizontal and vertical broken
lines on the quarter plane whose distribution is proved to be translation
invariant. This invariance follows from a reversibility property of the model.
This class of systems generalizes several classical processes of the same kind,
such as Hammersley's broken line processes involved in Last Passage Percolation
theory or such as the six-vertex model for some special sets of parameters. The
novelty comes here from the introduction of an intensity associated with each
line. The lines are initially generated by some spatially homogeneous weighted
Poisson Point Process and their evolutions (turn, split, coalescence,
annihilation) are ruled by a Markovian dynamic which preserves Kirchhoff's node
law for the line intensities at each intersection.
