Time for a crazy theory with no supporting evidence?
It seems a reasonably acceptable stance to state that the legendary Ragnar 'Lodbrok' Sigurdsson King of Denmark and Norway is a composite character based upon several historical persons, with this in mind it is possible that some of his sons are in fact brothers but not necessarily all of the same father.
Reading about "Ivar the Boneless" it seems equally acceptable to equate him with Ivarr Gudrodson, king of Dublin, although it is by no means a closed question.
My theory at this point is that the two Ivars are the same man and that as well as the brothers mentioned in the Irish Annals he is also brother of Ubbe and Halfdan as per the English Chronicles.
It is still possible that Ragnar is the father of these men and that they are brothers of Bjorn Jarnsida, etc but given the argument that Ivar of Dublin's father was Guðrøðr Ragnvaldsson/Gofraid, King of Lochlann it seems unlikely.
So my theory is Ivar follows his brother Amlaíb / Óláfr Gudrødsson, King of Dublin to Ireland from Norway and spends several years dominating Dublin/Ireland/Irish Sea. After about 10 -15 years of making good money other parties in Scandinavia (Denmark?) see a commercial opening in England and invite Ivar (or ask his permission as the dominant viking in the region?) to participate and together they invade England in 865.
Ivar spends a few years in England with his younger brothers(?) Ubbe and Halfdan but returns to Ireland in 870/1 to assist his older brother Olaf/Amlaib with the siege of Dumbarton in 870 before dying in Dublin in 873/4.
I did find an online copy of the Anglo Saxon Chronicles (from a link i think Harald had posted) in modern English and read the years covering the Great Heathen Army. I can't find the link again but im 90% certain that Ragnar's name never got a mention, Ivar and Halfdan are clearly mentioned as brothers but not as sons of Ragnar Lothbrok