  For every $\;b>1\;$ fixed, we explicitly construct $1$-dimensional families
of embedded constrained Willmore tori parametrized by their conformal class
$\;(a,b)$\; with $\; a \sim_b 0^+\;$ deforming the homogenous torus \;$f^b$ of
conformal class \;$(0,b).$ The variational vector field at $f^b$ is hereby
given by a non-trivial zero direction of a penalized Willmore stability
operator which we show to coincide with a double point of the corresponding
spectral curve. Further, we characterize for $b \sim 1$, $b \neq 1$ and $a
\sim_b 0^+$ the family obtained by opening the "smallest" double point on the
spectral curve which is heuristically the direction with the smallest increase
of Willmore energy at $f^b$. Indeed we show in \cite{HelNdi1} that these
candidates minimize the Willmore energy in their respective conformal class for
$b \sim 1$, $b \neq 1$ and $a \sim_b 0^+.$
