Axially asymmetric traveling fronts in balanced bistable reaction-diffusion equations

For a balanced  bistable reaction-diffusion equation, an axisymmetric traveling front has been well known. We prove that an axially asymmetric traveling front with any positive speed does exist  in a balanced bistable reaction-diffusion equation. Our method is as follows. We use a pyramidal traveling front for an unbalanced reaction-diffusion equation whose cross section has a major axis and a minor axis. Preserving the ratio of  the major axis and a minor axis to be a constant  and taking the balanced limit, we obtain a traveling front in a balanced bistable reaction-diffusion equation. The cross section of this traveling front is a compact set with a major axis and a minor axis when the constant ratio is not $1$.