  A matrix polynomial is a polynomial in a complex variable $\lambda$ with
coefficients in $n \times n$ complex matrices. The spectral curve of a matrix
polynomial $P(\lambda)$ is the curve $\{ (\lambda, \mu) \in \mathbb{C}^2 \mid
\mathrm{det}(P(\lambda) - \mu \cdot \mathrm{Id}) = 0\}$. The set of matrix
polynomials with a given spectral curve $C$ is known to be closely related to
the Jacobian of $C$, provided that $C$ is smooth. We extend this result to the
case when $C$ is an arbitrary nodal, possibly reducible, curve. In the latter
case the set of matrix polynomials with spectral curve $C$ turns out to be
naturally stratified into smooth pieces, each one being an open subset in a
certain generalized Jacobian. We give a description of this stratification in
terms of purely combinatorial data and describe the adjacency of strata. We
also make a conjecture on the relation between completely reducible matrix
polynomials and the canonical compactified Jacobian defined by V.Alexeev.
