I haven't even been able to FIND Jfrouter. Is this part of Joomla, Joomfish, or JoomSEF? I haven't been able to see it in any of the three. When I google it, I only see Joomla 1.5 examples that have it. We are running Joomla 1.0.

I'm sorry, I forgot you are using Joomla 1.0.
Are there all the numbers present for the URLs? For example you've posted:
en/speech-language-pathology/english-for-new-canadians-11.html

Are there URLs with all the numbers present from -2.html up to -11.html, or only the one with -11.html? Could you post more URLs that differ ONLY in the number? Or could you send me an access to your administration area to Tato emailová adresa je chráněna před spamboty, abyste ji viděli, povolte JavaScript so I can check this myself?

Best regards,
ARTIO Support Team

en/search-11.html index.php?option=com_search&lang=en&limit=30&limitstart=150&Itemid=99999999
en/search-12.html index.php?option=com_search&lang=en&limit=30&limitstart=30&Itemid=99999999
en/search-494.html index.php?option=com_search&lang=en&limit=30&limitstart=90&Itemid=99999999
en/search-5.html index.php?option=com_search&lang=en&limit=30&limitstart=0&Itemid=99999999
en/search-51.html index.php?option=com_search&lang=en&limit=30&limitstart=60&Itemid=99999999
en/search-548.html index.php?option=com_search&lang=en&limit=30&limitstart=120&Itemid=99999999
en/search-649.html index.php?option=com_search&lang=en&limit=50&limitstart=50&Itemid=99999999
en/search-654.html index.php?option=com_search&lang=en&limit=50&limitstart=0&Itemid=99999999
en/search-668.html index.php?option=com_search&lang=en&limit=50&limitstart=100&Itemid=99999999

Hello,

all these URLs seem to be coming from com_search component.
The strange thing is that the URLs (origina) do not include any keyword or search phrase.

You have 2 options:
1) adjust the search extension settings (e.g. define limit and limitstart as non-sef variables so they will not be included in the URL) and find our why the keyword is not included in the URL (may e.g. have relation to search module used)
or
2) disable SEF for search component (or switch it to "nocache" SEF URLs type) so they will not be stored to DB
(having SEF URLs for search results pages does not help much anyway)

The problem is not with the search Component. If you read back, you'll see that I gave MANY examples, of a wide array of pages. However, I was asked to give a list of examples of the same page with multiple SEF URLs.

This happens across the entire site, not just for search.

The problem is clear: The same pages are getting multiple SEF Urls because of additional (optional) arguments after the fact.

I've seen one other forum thread where the user solved the problem themselves by editing the core, but that's a ridiculous expectation on the users.

Hello,

sorry, I did not read the whole thread carefully. Anyway I think there are multiple problems mixed together.

1) First problem is the numbered URLs - e.g. the search ones. This was answered in the previous post.
2) Second is e.g. the different values used for language codes - in the search examples, there is "en" used, in the previous post, there is "english".

These does not seem to be neither default Joomla or JoomFish URLs as they should be consisten.
Aren't you using some special customizations or add-ons that would be modyfying the default URLs?

Also, I David asked you to send all the stored URLs like
en/speech-language-pathology/english-for-new-canadians-11.html
so also how do look the ...10.html, ...9.html, ...

Furthermore, in the begining you write you are using JoomSEF 2.2.6, please note that the latest version is 2.3.4 - you should always work with the latest version.
So, first of all (and btw. according forum rules), you should upgrade to the latest version!