Necessary and sufficient conditions for the existence of certain quadratic invariants under a phylogenetic tree

Invariants are functions of the probabilities of state configurations among lineages, with expected values equal to zero under certain phylogenies. For two-state sequences, the existence of certain quadratic invariants requires a symmetric substitution model. For sequences with more than two states, the necessary condition for the existence of certain quadratic invariants in terms of independent events is much stronger than symmetry. For DNA sequences, only three parameters are allowed in the substitution model, which includes Kimura's two-parameter model as a special case.