  We observe that general reducible curves in sufficiently positive linear
systems on K3 surfaces are of a form that generalises Kodaira's classification
of singular elliptic fibres and thus call them extended ADE curves.
  On such a curve $C$, we describe a compactified Jacobian and show that its
components reflect the intersection graph of $C$. This extends known results
when $C$ is reduced, but new difficulties arise when $C$ is non-reduced. As an
application, we get an explicit description of general singular fibres of
certain Lagrangian fibrations of Beauville-Mukai type.
