Before revision:  We derive some necessary and sufficient first order conditions for optimality and we characterize the optimal solution of the investment problem in terms of the "base capacity " process, i.e. the unique solution of a Representation Problem \`a la Bank-El Karoui . Under Markovian assumption, we show that the base capacity is in fact deterministic and coincides with the free boundary of the  optimal stopping problem naturally associated to the singular control one.
After revision:  We derive some necessary and sufficient first order conditions for optimality and we characterize the optimal solution of the investment problem in terms of the "base capacity " process, i.e. the unique solution of a representation problem in the spirit of Bank and El Karoui (2004). We show that the base capacity is in fact deterministic and coincides with the free boundary of the optimal stopping problem naturally associated to the singular control one.