Surfaces containing two circles through each point

We find all analytic surfaces in space ${\mathbb{R}}^3$ such that through each point of the surface one can draw two transversal circular arcs fully contained in the surface. The problem of finding such surfaces traces back to the works of Darboux from XIXth century. We prove that such a surface is an image of a subset of one of the following sets under some composition of inversions:

• the set $\{p+q:p\in \alpha ,q\in \beta \}$, where $\alpha ,\beta$ are two circles in ${\mathbb{R}}^3$;
• the set ${,2\frac{[p \times q]}{|p+q|^2}:p\in\alpha,q\in\beta,p+q\ne 0,}$,where$\alpha,\beta$aretwocirclesin${S}^2$;
• the set $\{(x,y,z):Q(x,y,z,x^2+y^2+z^2)=0\}$, where $Q\in \mathbb{R}[x,y,z,t]$ has degree $2$ or $1$. The proof uses a new factorization technique for quaternionic polynomials.