AbstractDarboux Wronskian formulas allow us to construct Darboux transformations, but Laplace transformations, which are Darboux transformations of order one, cannot be represented this way. It has been a long-standing problem to discover what other exceptions exist. In our previous work we proved that among transformations of total order one there are no other exceptions. Here we prove that for transformations of total order two there are no exceptions at all. We also obtain a simple explicit invariant description of all possible Darboux transformations of total order two.