**ِDefinition (1):** Call an inner model M "rigid" if there are no non-trivial elementary embeddings $j: M\longrightarrow M$.

It is possible that $L$ is not rigid, while the problem is open for $HOD$. But there is an important difference between these two models; $L$ is "absolute" while $HOD$ is not. By absoluteness I mean:

**Definition (2):** An inner model $M$ of $ZFC$ is "absolute" iff for any inner model $N$ of $ZFC$ containing $M$ we have "$M$ in the sense of $N$ is equal to $M$".

Also consider the following definition:

**Definition (3):** An inner model $M$ of $ZFC$ is "forcing absolute" iff for any set forcing extension $W$ of $V$ we have "$M$ in the sense of $W$ is equal to $M$".

Now my questions are about the possible role of absoluteness of an inner model of $ZFC$ in consistency of its "non-rigidity assumption".

**Question (1):** Suppose $M\neq L$ is a definable inner model of $ZFC$ which is absolute. Is it possible that $M$ is not rigid?

**Question (2):** Suppose $M\neq L$ is a definable inner model of $ZFC$ which is forcing absolute. Is it possible that $M$ is not rigid?