  We construct embeddings of $\widehat{\mathfrak{sl}}_2$ in lattice vertex
algebras by composing the Wakimoto realization with the
Friedan-Martinec-Shenker bosonization. The Kac-Wakimoto hierarchy then gives
rise to two new hierarchies of integrable, non-autonomous, non-linear partial
differential equations. A new feature of our construction is that it works for
any value of the central element of $\widehat{\mathfrak{sl}}_2$; that is, the
level becomes a parameter in the equations.
