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

Integrability of SDYM Equations for the Moyal Bracket Lie Algebra
where is a complex parameter and are the charges
defined in the previous section. Then we easily get
Differentiating the first equation of (14) with respect to and the second
one with respect to then employing Eq. (14) once more one obtains
As (15) is satisfied for every it follows that
i.e. fulfills ME. Thus substituting the definition of and into
(14) and changing by the general solution we arrive at the linear
system for ME
Integrability condition of the system (17) is given by ME. The solution
is analytic in and is assumed to be of the form
where exp stands for the exponential in associative Moyal algebra,
i.e.
Analogously, we can consider another linear system for ME
As before, ME is the integrability condition of (18). The solution
has the Laurent series expansion
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