Let $X$ be a random variable with known distribution $N(\mu, \sigma)$. Let $Y$ be a count variable, i.e., it is a discrete random variable with known finite expectation and finite variance, which takes integer values in the range $[0, n]$, for finite $n$. Let their population correlation be $\rho > 0$.
What in general can we say about the sufficient or necessary conditions for $X$ to have the same conditional distribution at every value of $Y$, either actually or approximately or asymptotically? (Obviously, except for expected values of $X$, which must covary over $Y$.) That is, the same conditional distribution but with expectation varying over values of $Y$. In other words, what if any attributes of the respective univariate distributions, or their joint distribution, would I see and be like, "Ah, it has this attribute, therefore $X$ must be distributed about the same at every $Y$!" or "It doesn't have that attribute, so $X$ can't be distributed about the same at every $Y$!"
To be clear, I'm not saying the respective univariate distributions are unknown and I need to prove this is the case. I'm saying the univariate distributions and their correlation are known exactly. Perhaps it is even known a priori that $X$ must be distributed the same at every $Y$. I'm asking, what about either or both distributions implies this is the case?
Edit: I'll add that this is motivated by trying to understand circumstances under which the usual assumptions for OLS regression do not have to be assumed, but can be deduced--at least approximately--as mathematical consequences of the distributions of the variables themselves. It seems to me that, in the statistical context, these "assumptions" are framed as such because real, empirical variables can have any wacky relationship, so you can always find a technical counterexample to any proposed rule. But, in the mathematical context, you can actually guarantee that variables have certain properties. You can know, for example, that $Y$ is strictly increasing in $X$.