Generic rsa keys to be used for passwordless ssh

I have use case where the requirement is to have passwordless ssh between 3 servers, I can use below command to make it happen

ssh-keygen -t rsa -f /root/.ssh/id_rsa -q -P "";chmod 600 /root/.ssh/id_rsa;for host in $(awk -v self="$(hostname)" '$2!="localhost" && $2!=self {print $2}' /etc/hosts); do cat /root/.ssh/ | ssh -i /binary/IBM-IIS.pem ec2-user@$host "cat >> /home/ec2-user/authorized_keys"; ssh -i /binary/AWS.pem ec2-user@$host "sudo su -c 'cat /home/ec2-user/authorized_keys >> /root/.ssh/authorized_keys'"; done

However it requires each server to be up and running, so I can generate a key and then push it other server. What I am looking for is a specific public private key pair which is generated before hand and I use the encrypted values to be pushed in each server, more like they all have the same private and public key, which makes it independent from server to be up.