I am looking for an example of two random variables $X,Y$ such that
(a) $X,Y$ are not independent.
(b) At least one of $X,Y$ is not normal.
(c) $E(X|y)$ (expected value of $X$ given $Y=y$) is linear in $y$, i.e. of the form $a+by$, and $E(Y|x)$ is linear in $x$.
(d) The correlation coefficient $\rho\neq \pm 1$.