How to model equality in Integer Linear Programming How to implement v=(a==b) using Linear Programming?
$$
v=
\begin{cases}
True, a=b\\
False, a≠b\\
\end{cases}
$$

Until now I tried the big M-Method.

To show a≤b:
$$a-b+Mv≤M$$
$$-a+b-Mv≤-1$$

To show b≤a:
$$-a+b+Mv≤M$$
$$a-b-Mv≤-1$$

When they are equal everything is okay, but when a≠b it doesn't work.
Use Cast to boolean, for integer linear programming, setting $x=a-b$ and $y=v$.  This only works if you have a constant upper bound on $|a-b|$.  Otherwise, I don't know how to express it in an ILP.

See also Express boolean logic operations in zero-one integer linear programming (ILP), Boolean variable true iff equation is satisfied in ILP, Boolean variable that captures whether an inequality holds.