  Gravitational and electromagnetic (EM) field of the Dirac electron is
described by the Kerr-Newman (KN) solution. We elaborate a regular source of
the KN solution which satisfies the requirement of flat space-time inside the
source and realization of the exact KN solution outside the source. This
requirement removes conflict between gravity and quantum theory and determines
many details of the source structure. In particular, we obtain that the KN
source should forms a gravitating bag model, similar to the known MIT and SLAC
bag models. As opposite to the known bag models, the self-interacting Higgs
field should be confined inside the bag, while outside the bag the gauge
symmetry should be unbroken to provide the external KN gravity. We show that
twistorial structure of the Kerr geometry (the Kerr theorem) determines
structure of the Dirac equation, resulting in a variable mass term, which is
generated inside the bag through interaction with the confined Higgs field.
Similar to the other bag models, ellipsoidal deformations of the KN bag creates
a string-like structure of the dressed electron -- circular string positioned
along perimeter of the KN bag.
