  We study the ultradiscrete analogue of Lax pair proposed by Willox et al.
This "pair" is a max-plus linear system comprising four equations. Our starting
point is to treat this system as a combination of two max-plus eigenproblems,
with two additional constraints. Though infinite-dimensional, these two
eigenproblems can be treated by means of the "standard" max-plus spectral
theory. In particular, any solution to the system can be described as a
max-linear combination of fundamental eigenvectors associated with each
soliton. We then describe the operation of undressing using pairs of
fundamental eigenvectors. We also study the solvability of the complete system
of four equations as proposed by Willox et al.
