  Following a recently considered generalization of linear equations to
unordered data vectors, we perform a further generalization to ordered data
vectors. These generalized equations naturally appear in the analysis of vector
addition systems (or Petri nets) extended with ordered data. We show that
nonnegative-integer solvability of linear equations is computationally
equivalent (up to an exponential blowup) with the reachability problem for
(plain) vector addition systems. This high complexity is surprising, and
contrasts with NP-completeness for unordered data vectors. Also surprisingly,
we achieve polynomial time complexity of the solvability problem when the
nonnegative-integer restriction on solutions is dropped.
