BTW , talking about illegitimate daughters of Max, do you think this lady posing with him is one of them?
Well, I did some research on the photographer Franz Weisbrod. He worked in Frankfurt / Main, not in Munich. His photographs had the signature "F. Weisbrod, phot.", since the early 1870s you read "F. Weisbrod, Nachfolger" on the photos. The German word "Nachfolger" means "successor", so I guess since then his son or someone who assumed his photo atelier took the pictures.
To sum up: The photo of Max and this woman was obviously taken in Frankfurt before 1870. So I thought of family connections between Munich and Frankfurt. There is Mathilde of Hessen-Darmstadt, of course, but in my opinion it can't be her.
Although it would be great to have a pic of one of Maximilian's illegitimate daughters, I can't imagine that the duke took one of these daughters to an atelier in Frankfurt (why not Munich?) to take some photos ... He was a liberal mind, of course, but if he had taken his illegitimate daughters into public, I guess we would know much more about them now.
Besides, we have to keep in mind that Max was still married to Ludovica, sister and aunt of many kings and queens, and that he was the father of the Empress of Austria. I guess, in the 19th century there would have been a public outcry if Max had appeared somewhere with his illegitimate daughter. No chance.