  Let X be an abelian scheme over a scheme B. The Fourier--Mukai transform
gives an equivalence between the derived category of X and the derived category
of the dual abelian scheme. We partially extend this to certain schemes X over
B (which we call degenerate abelian schemes) whose generic fiber is an abelian
variety, while special fibers are singular.
  Our main result provides a fully faithful functor from a twist of the derived
category of Pic$^\tau$(X/B) to the derived category of X. Here Pic$^\tau$(X/B)
is the algebraic space classifying fiberwise numerically trivial line bundles.
  Next, we show that every algebraically integrable system gives rise to a
degenerate abelian scheme and discuss applications to Hitchin systems.
