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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 successin constructions of integrable hierarchies corresponding to classical ABCDaffine Lie algebras, while the KW method is more applicable, which can be evenused in exceptional EFG affine Lie algebras. But in KW construction, it isquite difficult to obtain Lax equations for the corresponding integrablehierarchies, while in DJKM construction, one can derive Lax structures for manyintegrable hierarchies. It is still an open problem for the derivation of Laxequations from bilinear equations in KW construction. Therefore if we can showthe equivalent DJKM construction for the integrable hierarchies derived by theKW 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 homogeneousrepresentations by using the language of the lattice vertex algebras.

Equivalence of two constructions for $\widehat{sl}_2$--integrable hierarchies