  We examine problem of existence of stationary random fields with linear
regressions and quadratic conditional variances, introduced by Bryc in
"Stationary random fields with linear regressions" (Annals of Probability 29,
No. 1, 504-519). Distributions of the fields are identified and almost complete
description of the possible sets of parameters defining the first two
conditional moments is given. This note almost solves Bryc's problem concerning
fields undetermined by moments - the only remaining set of parameters for which
the existence of Bryc's fields is unclear has Lebesgue measure zero.
