The standard way to solve polynomial eigenvalue problems $P(\la)x=0$ is to convert the matrix polynomial $P(\la)$ into a matrix pencil that preserves its spectral information-- a process known as linearization.

When $P(\la)$ is palindromic, the eigenvalues, elementary divisors, and minimal indices of $P(\la)$ have certain symmetries that can be lost when using the classical first and second companion linearizations for numerical computations, since these linearizations do not preserve the palindromic structure.

Recently new families of linearizations have been

introduced with the goal of f