  We extend the method of PST formulation to find a systematic way to
covariantize several non-covariant Lagrangians of self-dual gauge fields. We
derive in detail the necessary basic formulas which are used to prove the
existence of extra local symmetry that allows us to gauge fix the auxiliary
fields therein and non-covariant formulations are restored. We see that, the
extra local symmetry in the PST and PSST formulations, which describe the
covariant Lagrangians in the 6D decomposition of $6=1+5$ and $6=3+3$
respectively, can be expressed as a simple linear form in the field strength.
However, although in this paper we have found the covariant Lagrangians in the
other decomposition of $6=2+4$, the extra local symmetry of the gauge field
cannot be expressed as a simple linear form in the field strength. We present a
no-go theorem to prove this specific property. We also find other covariant
Lagrangians with more complex decomposition of spacetime.
