To bring two-dimensional elliptical equations in the two-dimensional
canonical form, you need to solve, say, the ordinary differential
equation
Note now that even if you take to be real, will be complex. And that means that you need to know what happens to the coefficients , , and when you depart the real -plane into the complex domain. That may be fine if you know the coefficients analytically, but otherwise it is a problem.
Assuming that you can solve the system, call the integration constant
. Assuming that it is a differentiable function of and
, it will satisfy
Now set