  Penkov with co-authors stdied several types of Lie algebra
$\mathfrak{gl}(\infty)$. Completely different new types of super versions of
$\mathfrak{gl}(\infty)$ are introduced in this paper: these are Lie
superalgebras of supermatrices infinite in all directions with non-zero
elements of each matrix occupying a region around the main diagonal bounded by
certain curves, not straight ligns. Several open questions related with further
study of the Lie superalgebras introduced and with their possible applications
to dynamical systems, such as KdV, are formulated.
