  In this paper, we study the elliptic Kashiwara-Vergne Lie Algebra
$\mathfrak{krv}$, which is a certain Lie subalgebra of the Lie algebra of
derivations of the free Lie algebra in two generators. It has a natural
bigrading, such that the Lie bracket is of bidegree $(-1,-1)$. After recalling
the graphical interpretation of this Lie algebra, we examine low degree
elements of $\mathfrak{krv}$. More precisely, we find that
$\mathfrak{krv}^{(2,j)}$ is one-dimensional for even $j$ and zero $j$ odd. We
also compute $\operatorname{dim}(\mathfrak{krv})^{(3,m)} =
\lfloor\frac{m-1}{2}\rfloor - \lfloor\frac{m-1}{3}\rfloor$. In particular, we
show that in those degrees there are no odd elements and also confirm Enriquez'
conjecture in those degrees.
