  Let $G$ be a connected reductive group, and let $X$ be an affine
$G$-spherical variety. We show that the classification of
$\mathbb{G}_{a}$-actions on $X$ normalized by $G$ can be reduced to the
description of quasi-affine homogeneous spaces under the action of a
semi-direct product $\mathbb{G}_{a}\rtimes G$ with the following property. The
induced $G$-action is spherical and the complement of the open orbit is either
empty or a $G$-orbit of codimension one. These homogeneous spaces are
parametrized by a subset ${\rm Rt}(X)$ of the character lattice $\mathbb{X}(G)$
of $G$, which we call the set of Demazure roots of $X$. We give a complete
description of the set ${\rm Rt}(X)$ when $G$ is a semi-direct product of ${\rm
SL}_{2}$ and an algebraic torus; we show particularly that ${\rm Rt}(X)$ can be
obtained explicitly as the intersection of a finite union of polyhedra in
$\mathbb{Q}\otimes_{\mathbb{Z}}\mathbb{X}(G)$ and a sublattice of
$\mathbb{X}(G)$. We conjecture that ${\rm Rt}(X)$ can be described in a similar
combinatorial way for an arbitrary affine spherical variety $X$.
