The 4 incarnations of gaNapati
The gaNesha purANa has the 4 fold incarnations of gaNapati. They are:
1) vinAyaka in the kR^itayuga. 3 demons dhUmrAkSha, narAntaka and devAntaka assume great power and spread terror. So gaNapati is born of kAshyapa and his wife. He has a 10 handed form with lion vAhana and slays these asuras in a great battle at kAshi.
2)mayureshvara in the tretayuga. A demon called sindhu swallowed a pot of amR^ita he had obtained from rudra and became invincible. gaNapati assumed a form with 6 hands and captured mayUra a mighty son of vinAta and rode him to battle and slew sindhu after making him disgorge the amR^ita. It said that he then gave his peacock to his brother kumAra.
3)lambodara in the dvAparayuga. This was for the killing of the demon sindhura who arose from the yawn of brahma. brahma graced him with many boons and armed with these he spread terror. gaNapati was born of vyAsa the vAsiShtHa and captured a terrible rat that was wreaking havoc in his house. Then riding that mouse and emerging in a 4 handed form he slew sindhura in a deadly encounter.
4)dhUmraketu is the incarnation of the future in the kaliyuga. Here gaNapati is said to assume a 2 handed form riding a horse for the destruction of the mlecchas and dasyus who would be ruining the world and restoring it to the Aryas.
The incarnations of gaNapati in the G.P are a suggestive of a relatively late work, as they clearly have borrowed from the older layer of Pauranic myths. So the GP is clearly a second order purana resting on the older motifs of the primary purANas. The puNyakSetras of some of these myths are associated with Maharashtra and Gujarat supporting a later regional character of this stream of the gANapatya sect. This is similar to the pattern observed in the populist kaumara sect in south. Also note the methodlogical parallels of Pauranic reworking in the kaumara tract “tamil skanda purANaM” and the G.P.