  Four-dimensional Kerr-Schild geometry contains two stringy structures. The
first one is the closed string formed by the Kerr singular ring, and the second
one is an open complex string with was obtained in the complex structure of the
Kerr-Schild geometry. The real and complex Kerr strings form together a
membrane source of the over-rotating Kerr-Newman solution without horizon, $a
=J/m >> m .$ It has also been obtained recently that the principal null
congruence of the Kerr geometry, induced by the complex Kerr string, is
determined by the Kerr theorem as a quartic in the projective twistor space,
which corresponds to embedding of the Calabi-Yau twofold in the bulk of the
Kerr geometry. In this paper we describe this embedding in details and show
that the four folds of the twistorial K3 surface represent an analytic
extension of the Kerr congruence created by antipodal involution.
