Ok, this is how I'm interpreting your question: If you have a complex-analytic function then the families are orthogonal to the families . And if you're given a harmonic function then you can find it's complex conjugate so that the function is analytic and so the families are orthogonal to the families .

Your function is not harmonic so you can't use the CR equations to find the families orthogonal to it. In your exercise, you're given which is harmonic so that the CR equations could be used to find the orthogonal families to it. However, you can still find the orthogonal famililies to and it turns out to be an interesting problem in DEs: differentiate throughout to get:

and therefore the orthogonal families satisfy:

which you can solve by finding an integrating factor which I think is .

. . . did it kinda' quick. Work out the bugs if any ok.