  We study the relationship between singularities of finite-dimensional
integrable systems and singularities of the corresponding spectral curves. For
the large class of integrable systems on matrix polynomials, which is a general
framework for various multidimensional spinning tops, as well as Beauville
systems, we prove that if the spectral curve is nodal, then all singularities
on the corresponding fiber of the system are non-degenerate. We also show that
the type of a non-degenerate singularity can be read off from the behavior of
double points on the spectral curve under an appropriately defined
antiholomorphic involution. Our analysis is based on linearization of
integrable flows on generalized Jacobian varieties, as well as on the
possibility to express the eigenvalues of an integrable vector field linearized
at a singular point in terms of residues of certain meromorphic differentials.
