Determine all polynomials $P(x)$ with integer coefficients such that $P(x)=P(y)$ has infinitely many integer solutions in integer $x$ and $y$ with $x \neq y$.

Choose $P(x)=a_n(x-k)^{2n}+a_{n-1}(x-k)^{2n-2}+...+a_0$, with integers $k,n,a_n,a_{n-1}, ...,a_0$ Then $P(x)=P(2k-x)$ for every integer $x \neq k$, thus satisfy the condition.

Now, I have a gut feeling that if $P(x)=P(y)$ and $x+y=M$, then for every integer $x$, $P(x)=P(M-x)$, but I cannot analyze any futher.

So my question is: Are there any other types of polynomials $P(x)$ that satisfy the orange question above?

(Any answers or comments will be appreciated!)