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84 EXACT SOLUTIONS AND SCALAR FIELDS IN GRAVITY
where and are independent of In the gauge (18) one
gets
Finally, substituting (32) into (31) we obtain the linear system (17) and
inserting (31) under (32) into (30) one recovers the linear system (18).
But, as we know ME is the integrability condition of these systems. Thus
one arrives at the twistor construction for ME. [Remark : The factorization
(23) is gauge invariant i.e., the transformation
does not change the function Consequently,
one can choose the gauge such that is of the form (18)]
5. THE CASE OF
In previous sections we dealt with the analytic in solution of ME.
Now consider the case that it might be singular in but its pole is
of rank not grater than so assume solution of the form
Thus defining new function
In terms of which the ME might be written as:
In an explicit form (compare with Eq. (4).)
Hence, equalizing to 0 each term in power of one obtains: