**1.** Let $X$ be a smooth irreducible $\Bbb C$-variety,
on which the algebraic $\Bbb C$-group $G={\bf G}_{a,{\Bbb C}}$
(the additive group) acts freely on the right:
$$ X\times _{\Bbb C} G\to X,\quad (x,g)\mapsto x\cdot g.$$
Assume that there exists a surjective morphism
onto a smooth $\Bbb C$-variety $Y$
\begin{equation}\label{e:*}
\varphi\colon X\to Y \tag{$*$}
\end{equation}
whose fibres are the orbits of $G$ in $X$.
Then the morphism $\varphi$ is smooth, from which one can deduce that $\varphi$
induces a locally trivial fibre bundle (in the usual topology) of $C^\infty$-manifolds
$$\varphi\colon X(\Bbb C)\to Y(\Bbb C).$$

Question 1.Does it follow that $(*)$ is locally trivial in theflat topology, that is, a $Y$-torsorunder $G$? In other words, is the morphism $$ X\times_Y G\to X\times_Y X,\quad (x,g)\mapsto (x,x\cdot g) $$ an isomorphism of $\Bbb C$-varieties?

**2.** Assume that $(*)$ is a torsor.
Since $H^1({\Bbb C}(Y),G)=\{1\}$, we know that $(*)$ admits a rational section.

Question 2.Does $(*)$ admit aregularsection? In other words, does there exist a regular map (morphism) $s\colon Y\to X$ such that $\varphi\circ s={\rm id}_Y\,$?