Surfaces containing two isotropic circles through each point

We prove (under some technical assumptions) that each surface in R3 containing two arcs of parabolas with axes parallel to Oz through point has a parametrization (P(u,v)R(u,v),Q(u,v)R(u,v),Z(u,v)R2(u,v)) for P,Q,R,Z∈R[u,v] such P,Q,R have degree at most 1 u and v, Z 2 v. The proof is based on the observation one can consider parabola vertical axis as an isotropic circle; this allows us use methods recent work by M. Skopenkov R. Krasauskas which all surfaces Euclidean circles are classified. Such approach also find similar arbitrary same assumptions). Finally, we get results concerning top view (the projection along axis) question.