Virasoro action on pseudo-differential symbols and (Noncommutative) supersymmetric peakon type integrable systems

Using Grozman's formalism of invariant differential operators we demonstrate the derivation of N=2 Camassa-Holm equation from the action of Vect(S ^1|2) on the space of pseudo-differential symbols. We also use generalized logarithmic 2-cocycles to derive N=2 super KdV equations. We show this method is equally effective to derive Camassa-Holm family of equations and these system of equations can also be interpreted as geodesic flows on the Bott-Virasoro group with respect to right invariant H ¹-metric. In the second half of the paper we focus on the derivations of the fermionic extension of a new peakon type systems. This new one-parameter family of N=1 super peakon type equations, known as N=1 super b-field equations, are derived from the action of Vect(S ^1|1) on tensor densities of arbitrary weights. Finally, using the formal Moyal deformed action of Vect(S ^1|1) on the space of Pseudo-differential symbols to derive the noncommutative analogues of N=1 super b-field equations.