Bringing cats to Jordan (from US) (Amman)

We're bringing two cats with us to Amman. I called the Jordanian embassy in the US and they said they only need a health certificate. When I called Royal Jordanian, however, the agent said they also need a "travel certificate" or "pet passport" but he did not really know what this was.

Anyone have experience/tips for bringing pets in from the US? I can't seem to find a US "pet passport" though something like this seems to exist in the EU.

Thanks in advance,
Sami