Quaternion analyticity and conformally Kählerian structure in Euclidean gravity

Starting from the fact that the d=4 Euclidean flat spacetime is conformally related to the Kähler manifold H²×S², we show the Euclidean Schwarzschild metric to be conformally related to another Kähler manifold M²×S² with M² being conformal to H² in two dimensions. Both metrics which are conformally Kählerian, are form-invariant under the infinite parameter Fueter group, the Euclidean counterpart of Milne's group of clock regraduation. The associated Einstein's equations translate into Fueter's quaternionic analyticity. The latter leads to an infinite number of local continuity equations.