Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

80 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
where Simple calculation gives
the conservation law
Hence, there exists function such that
Thus we are led to an iteration procedure. Given the nth currents
and satisfying the equation for one defines
the function such that and Then
the currents are defined by
and we get the following conservation law
Thus one arrives at an infinite number of nonlocal conservation laws for
The term nonlocal stands for the fact that the currents
and are defined by integration and differentiation of lower
currents. For the nth charge we get
So the (n+l)th currents read
[The natural question arises what the relations are between the above
conservations laws and the ones obtained by C.P.Boyer and
[22, 23] for heavenly equation in 4 dimensions.]
3. LINEAR SYSTEMS FOR ME
Here we show how one can adopt some results obtained for SDYM
equations (L.L.Chau et al [20], L.L.Chau [21]) to get the linear systems
connected with ME. Define