Describe **all** the invariant 2-dimensional subspaces of $\mathbb{C}^4$ (or $\mathbb{R}^4$) of the nilpotent map
$$
N = \begin{pmatrix}
0 & 1 & & \\
0 & 0 & & \\
& & 0 & 1 \\
& & 0 & 0 \\
\end{pmatrix}
$$

Equivalently, describe **all** fixed points of the unipotent map $u = \exp(N)$ acting on the grassmanian of planes $\mathrm{Gr}(2,4)$ of $\mathbb{C}^4$ (or $\mathbb{R}^4$).

This seems like an easy Linear Algebra problem, but I don´t see how to solve it by easy arguments. We have, for example, the invariant planes $[e_1,e_2]$, $[e_3,e_4]$ and $[e_1,e_3]$ spanned by the standard basis. But there are many more, in fact, a *connected continuum* of them!

This is because this is an example of a *partial Springer fiber* (perhaps the simplest nontrival example). Such a space can be defined in the more general context of reductive algebraic groups and is known to be connected (see, for example, Chapter 6 of the 1995 AMS book *Conjugacy Classes in Semisimple Algebraic Groups*, by Jim Humphreys). Altough there are many deep results about the topology of these spaces (they are paved by affines, their cohomology is related to representations of the Weyl group and so on) I have not been able to use these results to give a concrete description of this simple example.

I would like to list **all invariant planes** of $N$ and, if possible, see how they are glued together. Perhaps the theory and approach one uses to solve this example can be used to describe concretely other higher dimensional examples.

all of them, because any invariant subspace containing a vector $v$ shoud contain $Nv$. $\endgroup$