  Previously the DeTurck 'trick' has been used to render the stationary
Einstein's equation a well posed elliptic system that may be solved numerically
by geometric flow or directly. Whilst in the static case for pure gravity with
zero or negative cosmological constant there is a simple proof that solving the
modified "harmonic" Einstein's equation leads to a solution of the original
Einstein system - i.e. not a Ricci soliton - in the stationary case this
argument no longer works. Here we provide a new argument that extends the
static result to the case of stationary spacetimes that possess a "$t$-$\phi$"
reflection symmetry. Defining a "soliton charge" from the asymptotic behaviour
of the solution, we show that this quantity is always non-positive. Provided
asymptotic conditions are chosen such that this charge vanishes, then
stationary solitons cannot exist.
