Preserve old cookie expire Irule

Based on your iRule need to add the expire but needs to be an additional command.

when HTTP_RESPONSE { 
    set cookies [HTTP::cookie names] 
    look for a persistence cookie being set
    if { $cookies contains "JSESSIONID" } {
        rewrites all cookies being
        set foreach cookie_name $cookies {
            set cookie_value [HTTP::cookie $cookie_name]
            set cookie_expire [HTTP::cookie expires $cookie_name]
            HTTP::cookie remove $cookie_name 
            HTTP::cookie insert name $cookie_name value $cookie_value domain ".mysite.com.ar" path "/" version 0
            HTTP::cookie expires $cookie_name $cookie_expire absolute
            
        }
    }
}

However your iRule is replacing each cookie with the same cookie name and value, you can simple update the domain and path instead of remove and replace, this should not change the expires value set by the backend server:

when HTTP_RESPONSE { 
    set cookies [HTTP::cookie names] 
    look for a persistence cookie being set
    if { $cookies contains "JSESSIONID" } {
        rewrites all cookies being
        set foreach cookie_name $cookies {
            HTTP::cookie domain $cookie_name ".mysite.com.ar"
            HTTP::cookie path $cookie_name "/"
        }
    }
}