We reconsider the singular control problem with marginal stability of the closed loop system, when the transfer matrix from the input to the output can have linearly dependent columns, and zeros on the extended imaginary axis. We present a new theoretical (existence) result and a new numerical algorithm, based on finding an orthogonal transformation of the matrix pencil associated to the Euler-Lagrange differential equations into a block-triangular form. We present an application in linear quadratic control of descriptor systems, under the constraints of physical realizability of the control and impulse-free and marginally stable closed-loop system.