plotting – Projections of the 3-dimensional phase-space of a non-autonomous ODE system

Given classical system of ODE:

$begin{cases} dot{x}=g \ dot{g}=k cdot (-g+frac{df}{dx}) \ dot{h}=-h+frac{d^2f}{d^2x} end{cases}$

where $f = e^{-x^2}$

I am constructing a three-dimensional phase space as follows:

    f = Exp(-x^2); dfdx = D(f, {x, 1}); dfdx2 = D(f, {x, 2});

    Manipulate(
 VectorPlot3D({g, k (-g + dfdx), -h + dfdx2}, {x, -2, 2}, {g, -2, 
   2}, {h, -2, 2}, PlotLabel -> Row({"k = ", k})), {k, 1, 15})

My mathematica version is 12.0, so the StreamPlot3D command is not supported for me, but I hope my results are correct.

Questions:

  1. How to visualize the steady state point on this graph?
  2. How to get projections of $x-g$,$x-h$ and $g-h$ planes?
  3. Is it possible in an older version to build an analogue of the command StreamPlot3D.

Best regards to all specialists.