employ the fact that such static Lorentzian spacetimes, exterior to a horizon, may be analytically continued in time to smooth Euclidean geometries. The Lorentzian Killing vector field continues to a vector field which generates a $U(1)$ isometry, and vanishes only at the point that continues to the Lorentzian horizon. Thus the problem of finding static vacuum solutions without horizon, or exterior to a single horizon, is included in the general Ricci-flat smooth Euclidean problem. The heart of the paper is to provide an algorithm to find smooth Ricci-flat Euclidean geometries in a fully covariant manner by directly solving the Ricci-flatness condition.

We begin, in Section 2, by finding a general and covariant method for gauge-fixing that generalizes the harmonic-coordinate gauge condition. Specifically, we borrow from the study of Ricci-flow the so-called DeTurck trick [@MR697987], in which, using a fixed reference connection, the Ricci-flow equation is rendered strictly parabolic without changing its geometric content. In our context the DeTurck trick renders the Einstein equation into a strictly elliptic system of partial differential equations, which we refer to as the Einstein-DeTurck equation. In Section 3, we then consider two types of algorithms for solving this elliptic system.

The simplest algorithm is to 