Verb preposition for "be at odds"

I found "be at odds" in some examples and in each example the proposition -which is used for it- is different:

• They're at odds over the funding of the project.

• Her version of events was at odds with the police report.

• The liberal-left are at odds on Libya.

• EPA, Eco groups at odds in climate change case.

• A man's at odds to know his mind cause his mind is aught he has to know it with.

Does it make a difference which preposition comes after "at odds"?

There are a few expressions at work here:

• To be at odds
• To be at odds with something/someone
• To be at odds to do something

In these examples:

They're at odds over the funding of the project.
The liberal-left are at odds on Libya.
EPA, Eco groups at odds in climate change case.

The people involved are “at odds”, that is, in disagreement or conflict, with one another, and they're at odds over, on, in, or about some subject.

In this example, however:

Her version of events was at odds with the police report.

The disagreement is between “her version of events” and “the police report”.

A man's at odds to know his mind cause his mind is aught he has to know it with.

In this case, “be at odds to do” means the same as “be hard pressed to do”; basically, it's difficult.

Here, "to be at odds" is a transitive idiomatic verb. Whatever follows it will obey normal grammatical convention. You could replace the term "at odds" with "conflicted" and get pretty much the same result:

• They're conflicted over the funding of the project.

• Her version of events conflicted with the police report.

So you can see that "at odds" isn't driving the choice of preposition as much as the sense of the action is.