  We study relative integral functors for singular schemes and characterise
those which preserve boundness and those which have integral right adjoints. We
prove that a relative integral functor is an equivalence if and only if its
restriction to every fibre is an equivalence. This allows us to construct a
non-trivial auto-equivalence of the derived category of an arbitrary genus one
fibration with no conditions on either the base or the total space and getting
rid of the usual assumption of irreducibility of the fibres. We also extend to
Cohen-Macaulay schemes the criterion of Bondal and Orlov for an integral
functor to be fully faithful in characteristic zero and give a different
criterion which is valid in arbitrary characteristic. Finally, we prove that
for projective schemes both the Cohen-Macaulay and the Gorenstein conditions
are invariant under Fourier-Mukai functors.
