  We study a family of equations defined on the space of tensor densities of
weight $\lambda$ on the circle and introduce two integrable PDE. One of the
equations turns out to be closely related to the inviscid Burgers equation
while the other has not been identified in any form before. We present their
Lax pair formulations and describe their bihamiltonian structures. We prove
local wellposedness of the corresponding Cauchy problem and include results on
blow-up as well as global existence of solutions. Moreover, we construct
"peakon" and "multi-peakon" solutions for all $\lambda \neq 0,1$, and
"shock-peakons" for $\lambda = 3$. We argue that there is a natural geometric
framework for these equations that includes other well-known integrable
equations and which is based on V. Arnold's approach to Euler equations on Lie
groups.
