A characterization of the Legendre involution on the class of generic frontals

We show that, under an additional mild assumption, on the class of generic frontals, any involution whose fixed point set is exactly the same as the fixed point set of the Legendre involution must be the Legendre involution (Theorem 2 in §1). Moreover, its natural complexification (Corollary 1 in §3) is simultaneously shown.