  The multi-field generalisation of the Bateman equation arises from
considerations of the continuation of String and Brane equations to the case
where the base space is of higher dimension than the target space. The complex
extension of this equation possesses a remarkably large invariance group, and
admits a very simple implicit form for its general solution, in addition to the
special case of holomorphic and anti-holomorphic explicit solutions. A class of
inequivalent Lagrangians for this equation is discovered.
