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!)

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!)

(If this question should be closed or off topic, please let me know. If this site cannot answer this question, let me know, I will delete this question immediately)