The additive group $\mathbb{Z}_n$ for any natural number $n$ forms an additive group which has 0 as identity element and 1 as generator.
Please provide me few examples of additive groups (with integer elements) like above with out element 0 and generator should n't be 1.
What I mean is if $\mathbb{A}$ is an additive group with $i$,$g$ as identity and generator elements of $\mathbb{A}$ then $i\ne 0$ and $0 \not \in \mathbb{A}$ and $g \ne 1$.
It seems to be implicit in the question that $\mathbb Z_n$ is an additive group of integers, i.e. a subgroup of $\mathbb Z$. This is not the case. $\mathbb Z_n=\mathbb Z/n\mathbb Z$ is the group of residue classes modulo $n$. Its elements are often represented by integers, but they are not integers (else $1$ would generate all of $\mathbb Z$), but classes of integers, e.g. the element of $\mathbb Z_n$ often represented by $1$ is $\{kn+1\mid k\in\mathbb Z\}$.
The only subgroups of $\mathbb Z$ are the groups $n\mathbb Z$ with $n\in\mathbb N$. They have $0$ as their identity, and more generally, any subgroup of a group inherits the identity element of the group. This is because it inherits the binary operation of the group, and the binary operation by definition only has that one identity element.
Thus, there are no groups of integers of the kind that you are looking for. You'd have to define a different group operation to get one (and then it would be questionable whether it still makes sense to call the objects "integers").