How do you identify the Primary Market Maker (the ax) for a given stock assuming there is one?

You can find the ax in a stock by looking up who does the most volumes.

Now there is the one that the various exchanges publish, but that doesn't really mean anything, or may not be as important as the Autex report which displays who does the most volume instituionally, normally, that is whoever underwrote the issue. But not always. It should be obvious why this should be given more weighting than an exchange report.

Also, it is utilized by the M and A departments of upstairs firms as a sales tool for obtaining business.
Traders are also bonused on having the lead in instituional volume in various sectors depending on firms
strengths.

There are probably several more reports out there now seeing there are so many exchanges and dark pools, but those were the two main ones.