Let \((S,P)\) be a surface singularity at \(P\). An arc on \((S,P)\) is an algebroid curve going through \(P\) on \(S\), given by formal power series in one variable. An arc is general on \((S,P)\) if its strict transform on the minimal desingularization is smooth and intersects the exceptional curve \(E\) transversally at a point lying on a Zariski dense open set of regular points of \(E\). A wedge on \((S,P)\) is a parametrization of \(S\) by formal power series in two variables; the wedge is centered at an arc if its parametrization comes from the wedge parametrization by simple substitution of two one-variable series. Generally speaking the authors show that a wedge centered at a general arc on a normal surface singularity \((S,P)\) lift to its minimal desingularization provided \((S,P)\) is a sandwiched singularity, i.e. it is the formal neighborhood of \(P\) on \(S\) obtained by blowing up a complete ideal \(I\) in the local ring of a closed point \(O\) on a non-singular algebraic surface \({\mathcal X}_0\) defined over an algebraically closed field \(k\).