Here another solutions: f(x,y)= x^m y^{1/m}, m=- 1, (1+iSqrt{3})/2, (1-iSqrt{3})/2.

Thank you very much, may be there exist an general solution?

V. Marikhin: I don't know. That is a good question.

So called semisymmetric quasigroups are solutions to this equation. Actually, they are defined by the axiom identical to the equation.

A quasigroup is a groupoid (S;.) such that for all a,b,c in S there are unique x,y in S, such that x.b = c and a.y = c are true. Because of both existence and uniqueness of x and y , we can define inverse operations of .:

x = c/b (right division) and y = a\c (left division).

One may define duals of these operations:

b*a = a.b , c\\a = a\c , b//c = c/b

to get six parastrophes (there are other names) of the quasigroup .

The above equation characterizes quasigroups (S;f) which are equal to duals of their

left (but also right) division operations.

It is a folclore in quasigroup theory. If you are interested I can give you the proof.

Thank you Aleksandar. I will think about your idea.