Integrable supermanifolds and associated nonlinear equations

A construction of the author [1] is generalized to embeddings of supermanifolds $V_{2\mid 2}$ in an enveloping superspace equipped with the structure of a Lie superalgebra. The general treatment is illustrated by the example of the series $\text{sl}(n,n+1)$, in which the integrable supermanifolds are described by supersymmetric equations of the type of the two-dimensional Toda chain and, in particular, for $n=1$ the Liouville equation and the sine-Gordon equation.