Sparse singular systems have to be solved for instance with semi-definite Neumann-problems. Such systems \(Ax=b\) with a rank deficiency of one can be solved by fixing a \(x_ i\), deleting the corresponding row and column, adjusting the right hand side and solving the new system with a preconditioned conjugate gradient method. Alternatively one may solve \(Ax=b_ R\) with \(b_ R=b-{\mathcal P}_{N(A)}b\) with the same method, if the kernel N(A) is known explicitly. The author shows, that this method often is faster than the first one. Conditions for the existence of a nonsingular incomplete Cholesky decomposition are given. The results are illustrated by a numerical experiment.