  We give integrable quad equations which are multi-quadratic (degree-two)
counterparts of the well-known multi-affine (degree-one) equations classified
by Adler, Bobenko and Suris (ABS). These multi-quadratic equations define
multi-valued evolution from initial data, but our construction is based on the
hypothesis that discriminants of the defining polynomial factorise in a
particular way that allows to reformulate the equation as a single-valued
system. Such reformulation comes at the cost of introducing auxiliary (edge)
variables and augmenting the initial data. Like the multi-affine equations
listed by ABS, these new models are consistent in multidimensions. We clarify
their relationship with the ABS list by obtaining Backlund transformations
connecting all but the primary multi-quadratic model back to equations from the
multi-affine class.
