Let $B$ be a connected, solvable algebraic group (of dimension 2) acting on a projective variety $Y$ (of dimension $3$): for instance, let $B$ be a Borel subgroup of a reductive algebraic group $G$.

**Question1**: Is it always true that there is a $B$-invariant complete flag (i.e. codimension $1$, irreducible and $B$-invariant subvarieties), $Y \supset Y_1\supset Y_2 \supset Y_3$? For my purposes I can assume that there is a point in $Y$ with finite stabliser group and therefore can choose the orbit-closure of that point to be $Y_2$. How about $Y_1$? Is there always a point with $1$-dimensional stabiliser group of the $B$-action on the just described $Y_2$? Obviously there will always exist a $B$-fixed point $Y_3$, by Borel's fixed point theorem, if $Y_1$ is a projective variety on which $B$ acts.

**Question2**: Does this generalize to arbitrary dimension $n$ of $Y$ and any reductive algebraic group $G$ with a Borel subgroup $B$?