Sorry to hear abt your mother.My prayers are with you and your family during this time.
Not being related to your family, I can only give you lookups(all from ancestry.com) and suggestions. First of all, and the most important thing if you are wanting to solve this, is to order the death certificates. Matthias Julius died Oct 7 1917 in Delphi Carroll Co IN. His age was listed at 66. Also marriage licenses: (1)Mathias Julius married Kate Lynn Aug 15 1879 Carroll Co IN. (2) Matthias Julius married Laura B Hopkins Mar 7 1889 Carroll Co IN.
On the 1900 census, Mat states he either arrived or was naturalized in 1852.There is a record of the naturalization in "An Index to Indiana Naturalization Records".There is no clue as to where to find them except the Indiana Historical Society.His record is on pg 77 and source publication is 3434.There is also an image of a passport he got in 1900; gives a description of him; his birth date and place. But, no relatives listed.
On the 1900 census there are 2 sons listed. Charles Otto was born May 19 1877 according to his WWl draft card(Delphi Carroll Co). He is a clothier(same as Matthias on the census) and is married to Effie May. He is med build & height with blue eyes and brown hair. On the 1910 census, he is one house away from Mat & Laura with no children.1920, he is in the same spot, Laura is widowed and Charles has a son, Robert C b 1923. 1930 census, there is still just Robert.SSDI has Robert listed as born Feb 5 1912 and died Nov 1892 Flora, Carroll Co., IN.
The only thing I have been able to find on the other son Harry was a possibility of being in CA 1920 married to a Gertrude with no children, his occ was listed as Sgt. US Army, which makes sense because I do not find a WWl draft card on him. Then I find a Harry same age b IN in Potter Co TX married to Jean and owning a cafe. No children with either of these listings.I do not find anyhthing else after that.
I think Matthias was left widowed with 3 children, one an infant possibly newborn. He did what he felt was best for all by letting a childless couple have the baby. That way he could take care of what he had and still feel the baby was being taken care of as well. Maybe some of those records will have a clue for you. If nothing else, you might look up the name Julius in the Carroll Co directory and call one of them.They may be able to pleasantly surprise you.
Let me know if you order the records and if they have information on them.From the death record on Matthias, you will probably get his parents' names. I would also order Charles', just to see if Mat was listed as father. Mystery as to why he was not on the 1880 census.