  We study a class of stationary Markov processes with marginal distributions
identifiable by moments such that every conditional moment of degree say $m$ is
a polynomial of degree at most $m\;\text{.}\;$ We show that then under some
additional, natural technical assumption there exists a family of orthogonal
polynomial martingales. More precisely we show that such a family of processes
is completely characterized by the sequence $\{(\alpha_{n} ,p_{n})\}_{n \geq
0}$ where $\alpha_{n}^{ \prime } s$ are some positive reals while $p_{n}^{
\prime } s$ are some monic orthogonal polynomials. Paper of Bakry\&Mazet(2003)
assures that under some additional mild technical conditions each such sequence
generates some stationary Markov process with polynomial regression.
  We single out two important subclasses of the considered class of Markov
processes. The class of harnesses that we characterize completely. The second
one constitutes of the processes that have independent regression property and
are stationary. Processes with independent regression property so to say
generalize ordinary Ornstein-Uhlenbeck (OU) processes or can also be understood
as time scale transformations of L{\'e}vy processes. We list several properties
of these processes. In particular we show that if these process are time scale
transforms of L{\'e}vy processes then they are not stationary unless we deal
with classical OU- process. Conversely, time scale transformations of
stationary processes with independent regression property are not L{\'e}vy
unless we deal with classical OU process.
