Our funding from the United States National Science Foundation has restrictions for travel that we must follow:
"Any air transportation to, from, between, or within a country other than the U.S. of persons or property, the expense of which will be assisted by NSF funding, must be performed by or under a code-sharing arrangement with a U.S.-flag air carrier if service provided by such a carrier is available. Tickets (or documentation for electronic tickets) must identify the U.S. flag air carrier's designator code and flight number."
In sum, the flight numbers must have the code from the U.S. airline (e.g., "UA" for United Airlines, "DL" for Delta, etc.) in front of each flight. Then it doesn't matter to whom the actual airplane belongs. If, for example, you fly with Lufthansa you will want to ensure that the flight numbers are preceded by the code "UA" and not the code "LH".
There are a few exceptions, for example when there is no U.S. airline code-shared flight out of your home city. In which case, the NSF says: "...a foreign-flag air carrier shall be used only to the nearest interchange point on a usually traveled route to connect with a U.S. flag air carrier." Please see the the full version of Air Travel Reimbursement rules for details.