We find all analytic surfaces in 3-dimensional Euclidean space such that through each point of the surface one can draw two transversal circular arcs fully contained in the surface. The search for 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:|p+q| 2 : p ∈ α, q ∈ β, p + q = 0 }, where α, β are two circles in S 2 ; -the set { (x, y, z) : Q(x, y, z, x 2 + y 2 + z 2 ) = 0 }, where Q ∈ R[x, y, z, t] has degree 2 or 1. The proof uses a new factorization technique for quaternionic polynomials.

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