It is not so much the airline issue you need to deal with in terms of whether they will transport the cat (that you would need to call ahead and work out with the airline and it would not be an issue).
The real issue you have is international travel with your animal... Each international country has its own requirements and, even if you are able to get the cat into Ecuador using the information from the site below, you are then going to have even more troubles bringing the cat back into the United States from Ecuador as that will even be more difficult.
Here is a site that shows all of the necessary steps to even take the cat to Ecuador (and they are extensive)
Once again, bringing the cat back into the United States once it has been in a foreign country will be even more difficult.
Is there anyone that you trust that could watch the cat while you are away?