  We study rank two locally-free Fourier-Mukai transforms on K3 surfaces and
show that they come in two distinct types according to whether the determinant
of a suitable twist of the kernel is positive or not. We show that a necessary
and sufficient condition on the existence of Fourier-Mukai transforms of rank 2
between the derived categories of K3 surfaces X and Y with negative twisted
determinant is that Y is isomorphic to X and there must exist a line bundle
with no cohomology. We use these results to prove that all reflexive K3
surfaces (including the degenerate ones) admit Fourier-Mukai transforms.
