What test can we perform on a vector field to determine if there exist vector field(s) that describe its inverse curl?
What do you mean by inverse curl? Are you saying you have the curl and now want to find the field or fields for it?
Are you looking for a physical test or a mathematical test?
Physically, you could place a small rotor in the flow and see what happens.
Mathematically you should know already ie what is the curl of a conservative field.
The curl operator is not injective, hence there is no unique solution for the inverse curl. However, I just want to know the method by which we determine whether an inverse curl vector field does exist, given an existing vector field.
yes. whether it could be the curl of another vector field