  We investigate second order additive invariants in elementary cellular
automata rules. Fundamental diagrams of rules which possess additive invariants
are either linear or exhibit singularities similar to singularities of rules
with first-order invariant. Only rules which have exactly one invariants
exhibit singularities. At the singularity, the current decays to its
equilibrium value as a power law $t^{\alpha}$, and the value of the exponent
$\alpha$ obtained from numerical simulations is very close to -1/2. This is in
agreements with values previously reported for number-conserving rules, and
leads to a conjecture that regardless of the order of the invariant, exponent
$\alpha$ seems to have a universal value of 1/2.
