As I recall, a long time ago, there was a procedure departing from Lugano which might address your question.
Taking-off to the NE required a right 180 to climb taking advantage of the valleys. The initial procedure was the same for EO and All Engines. However, there were contingencies for an engine failure during the climb which required a turnoff from the main climb route (SID?) depending on aircraft type, performance, and position. The procedure was applied according to altitude-range constraints where the standard routing would not assure adequate clearance over distant terrain.
I recall three option points. Two required a turn after engine failure to take advantage of the valleys to the SE.
The third, after passing the second option point was where the engine failure performance was sufficient to continue the SID.
As for approval, I don’t know. At that time the local Swiss authority were somewhat independent, approving operational procedure as required, based on the operators justification; similar to drift down routing.
I only saw local Crossair (‘Swiss’) charts, not the authoritative Swiss AIP; the implication was that the procedure was a special, limited to that operator only