  Let $K$ be an algebraically closed field. Let $G$ be a non-trivial connected
unipotent group, which acts effectively on an affine variety $X.$ Then every
non-empty component $R$ of the set of fixed points of $G$ is a $K$-uniruled
variety, i.e, there exists an affine cylinder $W\times K$ and a dominant,
generically-finite polynomial mapping $\phi:W\times K\rightarrow R.$ We show
also that if an arbitrary infinite algebraic group $G$ acts effectively on
$K^n$ and the set of fixed points contains a hypersurface $H$, then this
hypersurface is $K$-uniruled.
