How to add a cut to avoid previously found solution?

Question

I do have a MILP problem. The decision variables are Integer ones. Let's call them $x_i$ and i $\in \{0,1,2,3,...,N \}$ I do want to find all distinct solutiuons for this problem.

The problem is solved and the optimal solution is found as $x^*_i$. If the decision variables where binary ones I could add the following cut and resolve to find a distinct new solution

for i in range(1,N+1):
    if $x^*_i$>0:
       expr+=1-x_i
    else:
        expr+=x_i

add the cut as expr $\geq 1$ and resolve the problem

The Idea can be

for i in range(1,N+1):
    if $x^*_i$>0:
       expr+=|x^*_i-x_i|
    else:
        expr+=x_i

but it is not linear ! Any advice would be appreciated.