I recently faced the problem of calculating shortest paths in undirected graphs in the presence of negative edge weights; I could not find any applicable algorithms via online search.

Transforming the undirected graph to a directed one by replacing each edge $e_{ij}$ with a pair of antiparallel arcs $a_{ij}$ and $a_{ji}$ and then applying the Bellman-Ford algorithm is not an option, because egdes with negative weight correspond to a negative cycle consisting of a pair of antiparallel arcs.

Imposing the constraint, that the successor of a vertex must not be equal to its immediate predecessor doesn't rule out all negative cycles that contain a pair of antiparallel arcs, and apart from that, leaves open the question of which of the antiparallel arcs should be excluded.

After some experimenting I found this candidate transformation of a pair of antiparallel arcs (and thus for undirected edges):

That transformation also can't rule out antiparallel arcs from appearing in a partial shortest path constructed with the Bellman-Ford algorithm; it appears to me however, that calculating the shortest path via mincost flow reliably rules out antiparallel arcs from appearing in the calculated path (in the mincost flow version the capacity of every arc is set to $1$ and the cost is set to the arc-weight and using the wellknown flow constraints for the shortest path calculation in the mincost flow formulation).

Questions:

have graph transformation already been described, that rule out the simultanous appearing of antiparallel arcs in the same directed cycles, i.e. that make them mutually exclusive in their appearance?

can the problem of ruling out antiparallel arcs in shortest paths be solved with an appropriate graph-transformation and a classical shortest path algorithm, i.e. an algorithm that is not based on mincost flow?