  We consider a family, depending on a parameter, of multiplicative extensions
of an elliptic curve with complex multiplications. They form a 3-dimensional
variety $G$ which admits a dense set of special curves, known as Ribet curves,
which strictly contains the torsion curves. We show that an irreducible curve
$W$ in $G$ meets this set Zariski-densely only if $W$ lies in a fiber of the
family or is a translate of a Ribet curve by a multiplicative section. We
further deduce from this result a proof of the Zilber-Pink conjecture (over
number fields) for the mixed Shimura variety attached to the threefold $G$,
when the parameter space is the universal one.
