  In this article, we compute the motivic Igusa zeta function of a space
monomial curve that appears as the special fiber of an equisingular family
whose generic fiber is a complex plane branch. To this end, we determine the
irreducible components of the jet schemes of such a space monomial curve. This
approach does not only yield a closed formula for the motivic zeta function,
but also allows to determine its poles. We show that, while the family of the
jet schemes of the fibers is not flat, the number of poles of the motivic zeta
function associated with the space monomial curve is equal to the number of
poles of the motivic zeta function associated with a generic curve in the
family.
