I develop an application that sends push notifications to iOS devices. It is not important (~useless) in my case to count the number of notifications on the app icon badge, as they are not those kind of messages.
Is it allowed by Apple NOT to increment or show show the notification count on the bade when getting a push notification? Or is it compulsory by Apple design rules?
You are allowed to do whatever you want with the app icon. The icon is not a count of received notifications. Instead, it's typically a count of actionable things to do within the app (where an action may very well be to read the notification message).
If, when launching your app, the user is presented with a list of notifications they missed, then it's probably a good idea to badge the app. But from your description it sounds like that's not the case.
