  Building on results of Clemens and Kley, we find criteria for a continuous
family of curves in a nodal $K$-trivial threefold $Y_0$ to deform to a scheme
of finitely many smooth isolated curves in a general deformation $Y_t$ of
$Y_0$. As an application, we show the existence of smooth isolated curves of
bounded genera and unbounded degrees in Calabi-Yau complete intersections
threefolds.
