Jacobi matrix suggests a fixed plane while it is not? The Jacobi matrix for the system
$$\dot{\theta}=v$$
$$\dot{v}=-\sin(\theta)+\gamma$$
is in the form
$$\begin{bmatrix}
0 & 1\\
-\cos(\arcsin(\gamma)) & 0
\end{bmatrix}$$
Now for the case $\gamma=\pm1$ it becomes
$$\begin{bmatrix}
0 & 1\\
0 & 0
\end{bmatrix}$$
which has $\tau=0$ and $\Delta=0$ and corresponds to a non-isolated fixed point, in this case, a fixed plane.
But for this system, the fixed points are calculated to be at where
$$\sin(\theta)=1, v=0$$
which in the vector field of this system can be seen:

So which result is correct? If this is the case of nonlinear terms effect which can't be seen in the Jacobi matrix, then how can we have non-isolated fixed points at all?
 A: The fringe cases should be ignored for this analysis.
When you find the fixed points of a system of ODEs, the theory states that if all of the eigenvalues of the Jacobian at that fixed point are nonzero you can describe the behavior of the system as if it were a linear system, meaning we can identify it as stable, unstable, a saddle, a source, a sink, etc. without any fear that this will fall apart (the radius around this fixed point in which this is valid is still a difficult question).
The fringe cases occur whenever we have one of two cases: either a repeated eigenvalue or a zero eigenvalue. These cases imply strange behavior in the linear case. For example, in 2D linear systems a repeated eigenvalue either makes a degenerate node or a star node, and a zero eigenvalue makes nonisolated equilibrium points. Since this is only an approximation of the nonlinear system, we should think that extremely close to the fixed point these might be true, but once we escape a tiny distance away this geometry falls apart and the whole returns to the usual source/sink/saddle analysis. As such, it's not really fair to say that the system has nonisolated equilibria because this only makes sense in the limit, not in the large, not even in a neighborhood.
All that said, you can surmise that in a tiny region around this looks like a collection of nonisolated equilibria, even if it isn't perfect. If you squint enough you can kind of convince yourself that its ...almost true...
