  We classify the discriminantly separable polynomials of degree two in each of
three variables, defined by a property that all the discriminants as
polynomials of two variables are factorized as products of two polynomials of
one variable each. Our classification is based on the study of structures of
zeros of a polynomial component $P$ of a discriminant. This classification is
related to the classification of pencils of conics in a delicate way. We
establish a relationship between our classification and the classification of
integrable quad-equations which has been suggested recently by Adler, Bobenko,
and Suris.
