On Different Formulations of Chiral Bosons

It is shown, that recently constructed PST Lagrangians for chiral supergravities follow directly from earlier Kavalov-Mkrtchyan Lagrangians by an Ansatz for the $\theta $ tensor by expressing this in terms of the PST scalar. The susy algebra which included earlier $\alpha$-symmetry in the commutator of supersymmetry transformations, is now shown to include both PST symmetries, which arise from the single $\alpha$-symmetry term. The Lagrangian for the 5-brane is not described by this correspondence, and probably can be obtained from more general Lagrangians, posessing $\alpha$-symmetry.