  We derive a new equation for the optimal investment boundary of a general
irreversible investment problem under exponential L\'evy uncertainty. The
problem is set as an infinite time-horizon, two-dimensional degenerate singular
stochastic control problem. In line with the results recently obtained in a
diffusive setting, we show that the optimal boundary is intimately linked to
the unique optional solution of an appropriate Bank-El Karoui representation
problem. Such a relation and the Wiener Hopf factorization allow us to derive
an integral equation for the optimal investment boundary. In case the
underlying L\'evy process hits any real point with positive probability we show
that the integral equation for the investment boundary is uniquely satisfied by
the unique solution of another equation which is easier to handle. As a
remarkable by-product we prove the continuity of the optimal investment
boundary. The paper is concluded with explicit results for profit functions of
(i) Cobb-Douglas type and (ii) CES type. In the first case the function is
separable and in the second case non-separable.
