  In 70's A.A. Kirillov interpreted the stationary Schroedinger
(Sturm-Liouville) operator as an element of the dual space to the Virasoro
algebra, i.e., the nontrivial central extension of the Witt algebra. He
interpreted the KdV operator in terms of the stabilizer of the Schroedinger
operator. By studying the coadjoint representation of the simplest nontrivial
central extension of a simplest stringy superalgebra, the Neveu--Schwarz
superalgebra, Kirillov connected solutions of the KdV and Schroedinger
equations. We extend Kirillov's results and find all supersymmetric extension
of the Schroedinger and Korteweg-de Vries operators associated with the 12
distinguished stringy superalgebras. We also take into account the odd
parameters and the possibility for Time to be a $(1|1)$-dimensional
supermanifold. The superization of construction due to Khesin e.a.
(Drinfeld--Sokolov's reduction for the pseudodifferential operators) relates
the complex powers of the Schroedinger operators we describe to the superized
KdV-type hierarchies labeled by complex parameter. Our construction brings the
KdV-type equations directly in the Lax form guaranteeing their complete
integrability.
