  We investigate conditions for solvability and Malliavin differentiability of
backward stochastic differential equations driven by a L\'evy process. In
particular, we are interested in generators which satisfy a locally Lipschitz
condition in the $Z$ and $U$ variable. This includes settings of linear,
quadratic and exponential growths in those variables.
  Extending an idea of Cheridito and Nam to the jump setting and applying
comparison theorems for L\'evy-driven BSDEs, we show existence, uniqueness,
boundedness and Malliavin differentiability of a solution. The pivotal
assumption to obtain these results is a boundedness condition on the terminal
value $\xi$ and its Malliavin derivative $D\xi$.
  Furthermore, we extend existence and uniqueness theorems to cases where the
generator is not even locally Lipschitz in $U.$ BSDEs of the latter type find
use in exponential utility maximization.
