Parity splitting of a mean-field dynamo threshold

An antisymmetric perturbation can split the thresholds of two degenerate mean-field dynamo parity classes. At first order, project the perturbation equation onto the adjoint null mode using the Fredholm alternative. If multiplying the unperturbed complex mode by $i$ leaves the parameter-derivative matrix element unchanged but reverses the perturbation matrix element, the two first-order threshold shifts are equal and opposite.