At the risk of starting this post like a middle-school paper, I want to first define integrity...
Integrity is "the quality of being honest and having strong moral principles and moral uprightness". Great. But what is "moral"? Another peek at a dictionary shows that morality is "principles concerning the distinction between right and wrong and good and bad behavior".
Fine. So, integrity is about being honest and maintaining one's principles regarding good and bad behavior. Now that THAT'S solved...
In a divorce (or even post-divorce), it can be really difficult for some people to maintain their integrity. They may be so psychically injured that they resort to name-calling, rumor-spreading, underhanded financial scheming, turning kids against a parent, and any number of other questionable behaviors.
Those partners/ex-partners spend their integrity like they have an unlimited supply of it. They spend their integrity as if it's a currency that can re-grow on trees. The problem is that integrity doesn't grow on trees and can't be reclaimed without a great deal of reparation and humility.
Think about anyone you've known who has had a habit of (or even just one significant period of) lying, manipulating, wiggling out of trouble, disparaging someone else, spreading gossip... Then think about what your impression of that person is. Do you trust them? Do you want to work with them? Or are you wary and not interested in spending much time with them at all?
When one or both partners in a breakup spend their integrity, they lose the chance to work together authentically for the sake of their children (if there are any) or themselves. How can one parent trust the intentions of the other parent when all indications are that his or her integrity has been all used up? The reality is, they can't. That loss of trust then permeates all interactions.
However, preserving your integrity just so you can work with your ex isn't the only reason to hold on to it. You need to preserve your integrity for yourself -- so you can, quite frankly, live with yourself and sleep well at night. You may also want to be a positive role model for your kids or others who will look at how you handled yourself.
When I work with individuals or couples who are in relationship crisis, one of the first things I tell them is that we will work together to help them maintain their integrity through the process. It's critical for one's self-respect to at least try to do that. It's also critical to have a safe place to vent when your ex is spending their own integrity faster than seems possible--mostly so you can process it, but also so you don't spend some of your own trying to defend yourself.
If you'd like more information on how to maintain your integrity (or even get it back!) during a breakup, let me know--I'd be happy to help. Email me at firstname.lastname@example.org or check out my website at www.touchingtrees.com.