  In 1926, H. Busch formulated a theorem for one single charged particle moving
along a region with a longitudinal magnetic field [H. Busch, Berechnung der
Bahn von Kathodenstrahlen in axial symmetrischen electromagnetischen Felde, Z.
Phys. 81 (5) p. 974, (1926)]. The theorem relates particle angular momentum to
the amount of field lines being enclosed by the particle cyclotron motion. This
paper extends the theorem to many particles forming a beam without cylindrical
symmetry. A quantity being preserved is derived, which represents the sum of
difference of eigen-emittances, magnetic flux through the beam area, and beam
rms-vorticity multiplied by the magnetic flux. Tracking simulations and
analytical calculations using the generalized Courant-Snyder formalism confirm
the validity of the extended theorem. The new theorem has been applied for fast
modelling of experiments with electron and ion beams on transverse emittance
re-partitioning conducted at FERMILAB and at GSI.
