  Suppose $v(x,y):\mathbb C\rightarrow \mathbb R$ is an entire harmonic
polynomial with no critical points in the right half plane. Let $z_1,
z_2\in\mathbb C$ lie on a level set of $v$ , and assume ${\rm Re}(z_2)>{\rm
Re}(z_1)\geq0$. We give a necessary and sufficient condition, depending only on
algebraic properties of the polynomial $v$, for when there exists a smooth real
function $f$ whose graph $x+if(x)$ lies on a level curve of $v$ connecting
$z_1$ to $z_2$. Inspired by GIT, we construct a Kempf-Ness functional on an
appropriate function space, and prove the functional is bounded from below and
proper if and only if a such a graph exists. As an application, we find a
stability condition equivalent to the existence of a solution to the deformed
Hermitian-Yang-Mills equation on the family of projective bundles $
X_{r,m}:=\mathbb P(\mathcal O_{\mathbb P^m}\oplus \mathcal O_{\mathbb
P^m}(-1)^{\oplus (r+1)})$ with Calabi Symmetry.
