Derivation of Post-Newtonian (PN) expression for acceleration in Schwarzschild geometry The expression for the acceleration of a near-earth satellite as presented in the IERS Technical note is given by
\begin{equation} \label{eq:problemeq} \tag{1}
\frac{d^2\mathbf{r}}{dt^2} = \frac{GM_E}{c^2r^3} \left\{\left[2(\beta+\gamma)\frac{GM_E}{r} - \gamma \dot{\mathbf{r}} \cdot \dot{\mathbf{r}} \right] \mathbf{r} + 2(1+\gamma)(\mathbf{r}\cdot\dot{\mathbf{r}})\dot{\mathbf{r}} \right\}.
\end{equation}
We are working in the Parametrised Post-Newtonian formalism, hence the dimensionless constants $\beta,\gamma$.
From what I can gather from here and here this expression can be derived from the "Schwarzschild isotropic one-body point mass metric". 
Now, I know what the Schwarzschild metric looks like in isotropic coordinates but I can't see where Eq. (\ref{eq:problemeq}) comes from.
Another point to make is that in GR the dimensionless parameters $\beta,\gamma$ are equal to unity and when substituted above gives a well known formula for Schwarzschild precession which is given by
\begin{equation} \label{eq:problemeq2} \tag{1}
\frac{d^2\mathbf{r}}{dt^2} = \frac{GM_E}{c^2r^3} \left\{\left[4\frac{GM_E}{r} - \dot{\mathbf{r}} \cdot \dot{\mathbf{r}} \right] \mathbf{r} + 4(\mathbf{r}\cdot\dot{\mathbf{r}})\dot{\mathbf{r}} \right\}.
\end{equation} 
Again, I don't recall this either. 
Any suggestions?
 A: I see that you are referencing my old paper. Well in my paper I was referencing to the other of your two references, the book by JPL. There is an expression numbered 4-26 on page 4-19 in that book that reduces to the expression you wrote above in the case of one static large mass and one small "test-body". The derivation of expression 4-26 is provided on page 4-22 to page 4-24 in the same book by the Jet Propulsion Laboratory. The derivation is a bit beyond me I must say but still.
https://descanso.jpl.nasa.gov/monograph/series2/Descanso2_all.pdf
Edit I just wanted to add, in case someone arrives here looking for an expression, that the expression above only works well in the weak field limit. As is seen it approximates GR by introducing two velocity-dependent terms and one repulsive inverse cube term. This does not work well in the strong field regime. As the repulsive term gets stronger as you close in on the black hole, "bouncing" will occur. In the simulations below the green circle represents the Schwarzschild radius and the red circle the radius of the "innermost stable circular radius". As seen in this post, the post-Newtonian expansion is available also at the 3PN level, including more terms. Maybe that expression will work better in stronger fields.

