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I am puzzled as to how much information you actually have and what the sources are.
You quote Hugh, junior, as having a father Hugh and Mother Mary nee Murray. This suggests to me that you have the birth certificate of Hugh jr or his parent's marriage certificate. These being most likely Irish ones is there no reference to a Church, Parish or Registraion district? Does a Scottish death certificate gives detail of the deceased's parents?
When I looked at the 1881 Census I hope I found the right Hugh living with his Mother, Mary Shaw, Brother Robert and Sister, Esther Bunting. As the latter was born in Glasgow she seemed to be a good starter for locating her parent's names plus the place and date of their marriage. (Each element of this data was given in the case of my own Grandmother who was curiously born in Govan, Glasgow when the rest of her siblings were born in Sheffield). I also wondered if you had made use of the registered births of Hugh jr's step-siblings inorder to discover the maiden name of his Mother?
I haven't looked at the IGI to see if anything is there and the "British Isles Vital Records" did not bring anything to light.
So it would seem that Scottish General Records could, as they are better than the English counter part, be of help to you (www.scotlandspeople.gov.uk) but to acquire records you will have to pay! £6 for 30 pages worth. All the fee to be used within 168 consecutive hours.
A book "Irish and Scotch-Irish Ancestral Research", by Margaret D Falley, may also help.
As far as I am aware the Bunting Society does not have anything about Hugh jnr's family and ancestry in our records so anything you can offer would be welcome. Even if you personally descend from the Shaw line. We do have a Hugh born in Glasgow but to a different set of parents!
Good hunting. Michael.
Thanks for your advice. I did as you said and have now tracked them down to Lurgan. 1866 October 12th. Hugh Bunting and Mary Murray were married. Just finding it a nightmare to find any results on irish geneology!