Say we have the projective space $\mathbb P^2_{\mathbb R}$ = $\mathbb P(\mathbb R^3) \stackrel {\text{def.}}{=} \{\text{span(u)}\mid u\in\mathbb R^3\smallsetminus\{0\}\}$. Denote $[u]$ for an element of $\mathbb P^2_{\mathbb R}$, where $u\in\mathbb R^3$ (which can be thought of as en element of $\mathbb R^3 /\sim$, where $\sim$ is the "is proportional to" equivalence). Obviously $[u]=[\lambda u], \ \forall\lambda\in \mathbb R\smallsetminus\{0\}$. Also define, for $V\subset \mathbb R^3$, $V$ vector subspace, $[V]$ as $ \mathbb P(V)$, in the same way as before. A projective transformation in $\mathbb P^2_{\mathbb R}$ (along with being characterized in many many ways) is a bijective map $$f:\mathbb P^2_{\mathbb R}\to\mathbb P^2_{\mathbb R} $$ such that, for $[u]\in\mathbb P^2_{\mathbb R}, f([u]) = [\varphi(u)]$ (write $f = [\varphi]$), for some $\mathbb R^3\to\mathbb R^3$ vector-space isomorphism $\varphi$ (well defined, save multiplication by scalars).
For anyone who has understood my hurried definitions, here's my question:
Say we have $\varphi :\mathbb R^3\to\mathbb R^3$ with Jordan-form $$ \left( \begin{array}{ccc} \lambda & 1 & 0 \\ 0 & \lambda & 1 \\ 0 & 0 & \lambda \end{array} \right) $$ in (ordered) basis $\{u,v,w\}$. Consider the projective transformation $[\varphi]$. What can we say in regards to fixed points and invariant lines? Obviously $f([u]) = [\varphi(u)]] = [\lambda u] = [u]$, so $[u]$ is a fixed point, and $f([\alpha u+\beta v]) = [\alpha\varphi(u)+\beta\varphi(v)] = [\alpha\lambda u + \beta u + \beta\lambda v]$, meaning that the line $[\text{span}(u,v)]$ is invariant.
How do I know if I'm missing something? It doesn't look like theres any other invariant lines but how can I be sure? In general, what strategy does one use for this problem, where $\varphi$ has any other Jordan-form?