Stationary Markov chains with linear regressions

In Bryc (Ann. Probab., (1998), to appear), we determined one-dimensional distributions of a stationary field with linear regressions (1) and quadratic conditional variances (2) under a linear constraint (7) on the coefficients of the quadratic expression (3). In this paper, we show that for stationary Markov chains with linear regressions and quadratic conditional variances the coefficients of the quadratic expression are indeed tied by a linear constraint which can take only one of the two alternative forms (7), or (8).