By Lie's flag theorem: If $G$ is a connected solvable linear group in $Gl_n(\mathbb C)$, then $G$ stabilizes a full flag of complex subspaces $$\{0\}\subset V_1\subset\dots \subset V_n=\mathbb C^n$$ Now since $\mathbb CP^{n-1}=\mathbb C^{n-1}\sqcup\mathbb C^{n-2}\sqcup\dots \sqcup\mathbb C^1\sqcup\{\infty\}$, so can we deduce that if $G$ is solvable connected Lie group (linear in $Gl_n(\mathbb C)$) then $G$ stabilizes $$\{\infty\} \subset\mathbb CP^1\subset\mathbb CP^2\subset \dots\subset \mathbb CP^{n-1}$$
Is there any example on this situation?