  In the framework of Polynomial Eigenvalue Problems, most of the matrix
polynomials arising in applications are structured polynomials (namely
(skew-)symmetric, (skew-)Hermitian, (anti-)palindromic, or alternating). The
standard way to solve Polynomial Eigenvalue Problems is by means of
linearizations. The most frequently used linearizations belong to general
constructions, valid for all matrix polynomials of a fixed degree, known as
{\em companion linearizations}. It is well known, however, that is not possible
to construct companion linearizations that preserve any of the previous
structures for matrix polynomials of even degree. This motivates the search for
more general companion forms, in particular {\em companion $\ell$-ifications}.
In this paper, we present, for the first time, a family of (generalized)
companion $\ell$-ifications that preserve any of these structures, for matrix
polynomials of degree $k=(2d+1)\ell$. We also show how to construct sparse
$\ell$-ifications within this family. Finally, we prove that there are no
structured companion quadratifications for quartic matrix polynomials.
