We derive a new equation for the optimal investment boundary of a general

irreversible investment problem under exponential Lévy 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 Lévy process hits any point

in R 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.