  In this paper, we investigate the equivalence of Date--Jimbo--Kashiwara--Miwa
(DJKM) construction and Kac--Wakimoto (KW) construction for
$\widehat{sl}_2$--integrable hierarchies. DJKM method has gained great success
in constructions of integrable hierarchies corresponding to classical ABCD
affine Lie algebras, while the KW method is more applicable, which can be even
used in exceptional EFG affine Lie algebras. But in KW construction, it is
quite difficult to obtain Lax equations for the corresponding integrable
hierarchies, while in DJKM construction, one can derive Lax structures for many
integrable hierarchies. It is still an open problem for the derivation of Lax
equations from bilinear equations in KW construction. Therefore if we can show
the equivalent DJKM construction for the integrable hierarchies derived by the
KW construction, then it will be helpful to get corresponding Lax structures.
Here the equivalence of DJKM and KW methods is showed in the
$\widehat{sl}_2$--integrable hierarchy for principal and homogeneous
representations by using the language of the lattice vertex algebras.
