Boris Eremin was gifted with several aircraft.
possibly as many as 5 or 6 (can't seem to find where I read that)
sadly , I don't think the machine at Saratov represents any one machine that he flew
but is just a general representation.
(the Winter Yak-1b's seem to best photographed, perhaps they should have done one of these)
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You may be right, although a translated version of the airaces.narod.ru page that I linked above does describe in some detail (quite a bit of detail, actually) the history of this specific Yak-1b, even to its serial number, and relates this particular aircraft to its donor. If this description is correct, the Yak-1b at Saratov is an aircraft that was flown extensively by Eremin, over a relatively long period of time. It was taken out of service because of various kinds of damage which eventually made it unsafe to fly, and then (apparently after a period of sitting idle) taken to Saratov to be restored and put on display. Unfortunately the Babelfish translation doesn't have its own unique URL, so I can't link it, and a copy-and-paste of the whole text would make this an awfully long post!
Regarding the originality of the markings and camouflage, it does say the following, though:
"At the same time lower camera hatch was sealed, the equipment of cab is changed. From one restoration to the next painting machine and grant inscription (its for the first time changed during May 1943, after removing orthographical error in writing of the surname of donor) changed. On the cowling it was represented the 8 (number of biased on this Yak-1 enemy aircraft), then 14 (total number of biased machines on the grant "yaks" Of f. p. golovaty) asterisks. Thus, from the initial version of destroyer it remained sufficiently little. In all, with the resource 300 it is hour, aircraft flew 102 hours 25 minutes."
It's quite interesting, and I very much recommend you have a go at it with Babelfish. The translation is quite good and it can be easily understood, with a bit of imagination.
John