  This paper deals with magnetic equations of the type dH=J where the current J
is a delta-function on a brane worldvolume and H a p-form field strength. In
many situations in M-theory this equation needs to be solved for H in terms of
a potential. A standard universality class of solutions, involving
Dirac-branes, gives rise to strong intermediate singularities in H which in
many physically relevant cases lead to inconsistencies. In this paper we
present an alternative universality class of solutions for magnetic equations
in terms of Chern-kernels, and provide relevant applications, among which the
anomaly-free effective action for open M2-branes ending on M5-branes. The
unobservability of the Dirac-brane requires a Dirac quantization condition; we
show that the requirement of ``unobservability'' of the Chern-kernel leads in
M-theory to classical gravitational anomalies which cancel precisely their
quantum counterparts.
