= Solution
The tangent plane
$$
\boxed{x=1}
$$
cuts the hyperboloid in
$$
y^2=z^2,
$$
so its intersection is the pair of straight lines
$$
\gamma_\pm(t)=(1,\pm t,t).
$$
Their <accelerations> vanish, hence are normal to the surface, so both lines are geodesics. Their <tangent vectors> at their intersection $(1,0,0)$ are $(0,1,1)$ and $(0,-1,1)$, whose <inner product> is zero. They therefore intersect at a <right angle>, giving the second case of <plane-section geodesics of the unit one-sheet hyperboloid>.
There are also geodesics entirely contained in $z>0$. In the coordinates
$$
X(z,\phi)=\left(\sqrt{1+z^2}\cos\phi,
\sqrt{1+z^2}\sin\phi,z\right),
$$
the metric is
$$
\frac{1+2z^2}{1+z^2}\,dz^2+(1+z^2)\,d\phi^2.
$$
For a unit-speed geodesic, the <Clairaut first integral for a surface of revolution> is $c=(1+z^2)\dot\phi$, and the constant-speed equation becomes
$$
\dot z^2=\frac{1+z^2-c^2}{1+2z^2}.
$$
Choose $c>1$ and initial height $z_0=\sqrt{c^2-1}$. The geodesic initially tangent to the parallel has $\dot z=0$, and the displayed identity prevents it from entering $0<z<z_0$. Since the parallel at $z_0>0$ is not itself a geodesic, the curve turns there and otherwise has $z>z_0$. Hence it remains entirely in $z>0$; this is a <geodesic trapped in one half of the unit one-sheet hyperboloid>.
Finally, the waist
$$
\gamma(t)=(\cos t,\sin t,0)
$$
is a geodesic and is preserved setwise by every <isometry> of the hyperboloid. Indeed, its <Gaussian curvature> is
$$
K(z)=-\frac1{(1+2z^2)^2}.
$$
The value $K=-1$ occurs exactly at $z=0$. Since <Gaussian curvature> is intrinsic and therefore preserved by <isometries>, every isometry preserves the waist. Thus the answer to the final question is also yes, with the <isometry-invariant waist geodesic of the unit one-sheet hyperboloid> as an example.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2023/ib/paper-3-hyperboloid-geodesics.png]
{title=Plane-section geodesics on the unit one-sheet hyperboloid}
{description=Blue meridians are disjoint geodesics in the plane y equals zero, red ruling lines are geodesics meeting orthogonally in the tangent plane x equals one, and the purple waist is an isometry-invariant geodesic.}
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