  We propose and explore a "stationary 1+log" slicing condition for the
construction of solutions to Einstein's constraint equations. For stationary
spacetimes, these initial data will give a stationary foliation when evolved
with "moving puncture" gauge conditions that are often used in black hole
evolutions. The resulting slicing is time-independent and agrees with the
slicing generated by being dragged along a time-like Killing vector of the
spacetime. When these initial data are evolved with moving puncture gauge
conditions, numerical errors arising from coordinate evolution are minimized.
In the construction of initial data for binary black holes it is often assumed
that there exists an approximate helical Killing vector that generates the
binary's orbit. We show that, unfortunately, 1+log slices that are stationary
with respect to such a helical Killing vector cannot be asymptotically flat,
unless the spacetime possesses an additional axial Killing vector.
