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In both cases, x and y are real variables, while T, w are real-valued in case (i) but complex-valued in (ii). In the real case (i), 1/T is what is usually called an integrating factor for co. The solution of the problem in this case depends on the theory of ordinary differential equations. Part I will deal with this case. While the problem in the complex case (ii) has the same appearance as the problem in the real case, it is of a very different nature. Its solution depends on an elliptic system of partial differential equations. In fact, the problem is equivalent to the problem of conformalizing the Riemannian metric

On integrating factors and on conformal mappings