Notation:  ∂_k where k is a natural number differentiates a partially differentiable function with respect to its kth place. If the reader prefers the traditional notation he can easily translate back to it.
If W and Z are a finite paired node and branch-point of order 2 they are a solution of the simultaneous equations f(W,Z)=0 and ∂₁f(W,Z)=0. In this note it is assumed that other parameters are present though not shown. W and Z are by the local theory differentiable and therefore continuous functions of the other parameters wherever the determinant
|∂₁f   ∂₁∂₁f|
|∂₂f   ∂₂∂₁f|
does not vanish.  But ∂₁f =0 is given so the condition is that neither ∂₁∂₁f nor ∂₂f vanish.
It may be that more partial derivatives than the first with respect to the dependent variable vanish. Let ∂₁^hf  be the first which does not vanish. Then the singularity  is of  order h and remains so under variation  of  the  parameters  if these conditions are preserved.
If we are interested in just the topology of the branches we can with these safeguards give the parameters numerical values which make whatever calculations are needed as simple as possible.