Second order additive invariants in elementary cellular automata

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.