Clarence P. Burke (b. abt 1853) in Sweden as was his wife, Matilda C.   The Burkes were Protestant, Swedish.  Maybe a "J" was dropped at immigration and that the original name was Bjerke or Bjørke or Bjork?
Children:
Matilda C. Burke, 47, b. June 1856, Sweden Sweden Sweden (this is then Elmer's mother)
August V. Burke, son, 15, b. January 1884, Illinois Sweden Sweden (Elmer's older brother, born in Illinios)
Elmer C. Burke, son, 9, b. May 1891, Nebraska Sweden Sweden (Elmer himself - the birth year matches, born in Nebraska)
Maude Burke, dau., 5, b. July 1894, Nebraska Sweden Sweden (Elmer's younger sister, also born in Nebraska
Note: Clarence Elmer Burke (b. May 1891), aka Elmer C., C. Elmer, C. E. Burke.

1860:

1870:

1880 Census:
They do not appear on the 1880 US census, so Pete and Matilda may have married in Sweden, then immigrated to America.

1885 Census:
The family does not appear in the Nebraska state 1885 census, though there are other Swedish Burkes, none in Greeley county.

1880-1884: ?? ship manifests from this period

1900 NE Census:
Burke, Pete, 48, b. June 1857, Sweden Sweden Sweden, farmer (I'm thinking this was his father, Pete Burke or C.P. Burke)
Matilda C., 47, b. June 1856, Sweden Sweden Sweden (this is then Elmer's mother)
August V., son, 15, b. January 1884, Illinois Sweden Sweden (Elmer's older brother, born in Illinios)
Elmer C., son, 9, b. May 1891, Nebraska Sweden Sweden (Elmer himself - the birth year matches, born in Nebraska)
Maude, dau., 5, b. July 1894, Nebraska Sweden Sweden (Elmer's younger sister, also born in Nebraska
Ref: 1900 NE census, Clay Twp, Lynn County, Nebraska, District 38, page 121A: ln. 29, HH 134, Fam. 135.

1910 NE Census:

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Clarence Elmer Burke (b. May 1891)
Children:
Elaine E. Burke (b. abt 1916)
Lemoine E. Burke (b. abt 1918) aka "Bud"
Lillian T. Burke (b. abt 1920) (m.) Don Wheeler lived in Council Bluffs)
Martina E. Burke (b. abt 1921) (m.) Robert Rover lived in California)
Doris M. Burke (b. abt 1923) (m.) Jim Wood of Greeley, lived in AZ & CA
Lucette G. Burke (b. abt 1928) (m.)Gene Jensen of St. Ed)(must be Lucille deceased)

Cattle brands is CE (the E is sideways) is registered to Lemoine (Bud) or his wife Maybelle Burke (who live in
Greeley).

I found in the 1930 Federal Census these Greeley residents
Elaine E. Burke born abt. 1916 (Steve's grandmother, deceased, married at
15 or 16 to Patrick McManus, 3 children Ann, James E. & Jerry)
Lemoine E. Burke born abt 1918 (that's Uncle "Bud")
Lillian T. Burke born abt 1920 ( married Don Wheeler lived in Council
Bluffs)
Martina E. Burke born abt 1921(married to Robert Rover lives in California)
Doris M. Burke born abt 1923 (married Jim Wood from Greeley, lived in
Arizona now lives in California)
Lucette G. Burke born abt 1928 (that must have been Lucille?deceased,
married to Gene Jensen from St. Ed)
These are the siblings, Steve's grandmother and her younger brother and
younger sisters.

Next on the list must be the parents ???
Elmer C. Burke born abt. 1891 (so, unlike the brand, this says Elmer C not
C. Elmer)
Esther T. Burke born abt 1981 (where did she come from? maiden name?)
and then is the next name Elmer's father?
Clarence P. Burke born abt 1853
There are 3 more Burkes in Greeley County in the 1930 Federal Census, but
I'm thinking they're not related. The Burke's above were Protestant
Swedish and the names below are typically Catholic. Do you know if there
is any family connection? For that matter Burke is not a typical Swedish
name, I can't help but think a J was dropped at immigration and that the
original name was Bjerke or Bjørke or Bjork.

Eileen C. Burke born abt. 1916
Catherine E. Burke born abt 1880
Mary Frances Burke born abt 1870

I don't have access to The Pioneer History of Greeley County, but it is
referenced on a web site I found yesterday. There I found the following
Burkes.
C.P. Burke page 127 
(I assume that's Clarence P. Burke born 1854 Steve's great great
grandfather. Can you confirm? Where was he born, there weren't many
people in Nebraska in 1854?)
John Burke page 58
Mary Burke pages 35, 58, 173
(since she and John Burke are both mentioned on pg. 58 I assume they're
related to each other. But, are either of them related to us? Mary is not
a typical Swedish first name)
T.G. Burke page 123, 126
(page 126 is the page before 127, is there any connections?)
Thomas Burke page 35, 61
(related to Mary? - on page 35?
Thomas Burke page 166
From an 1884 list I found
Charles P.W. Burke Freeman Valley, Greeley County
James J. Burke, Freeman Valley, Greeley County
First question, where was Freeman Valley? I can find Leo Valley on an old
map, but I can't find Freeman Valley. And then the obvious question,
Charles PW cant be the same person as Clarence P can it?
M. Burke, O'Connor
Thomas G. Burke, O'Connor
Is this the same T.G. Burke found in The Pioneering History of Greeley
County? If it is then I assume it's another Burke family.

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James E. McManus Family

? McManus (m.) ? wife deceased
Children:
3 deceased

? McManus (m.) ?
Children:
James E. McManus (b. 1934) in Fish Creek township

James E. McManus (m.)
Children:
Patrick McManus

Patrick McManus (m.) Elaine E. Burke (b. abt 1916)
Children:
Ann McManus
James E. McManus
Jerry McManus

According to James E. McManus, the story goes that his grandfather McManus 
lost his wife and three children all in the same night, after that "the man 
was never worth a damn." Did they die of "consumption" - tuberculosis? What 
year was this?

From the name index to "The Pioneer History of Greeley County," I found
Anna McManus page 174
James McManus page 169
Are these the parents? Was Anna the wife who died? Was James Sarah's twin
brother? Were they once in the newspaper as Nebraska's oldest living
twins? Is this Patrick's father, James E.'s grandfather and Seve's great
grandfather?
Maggie McManus page 174
Is she one of the children who died? Neither Steve or his father have
ever heard of Maggie.
Mary McManus page 174
(Is this Aunt Mary Wood who married Orin? )
Miss Sarah page 133
(Is this another one of the children who died that night?)
Patrick page 62, 133
(I assume this is Steve's grandfather and James E.'s father, married to
Elaine Burke - he would have been born about 1910 and he died when Steve &
I were in 11th grade, I think-1972 or 1973)
Sarah page 169
(Is this the third child who died that night?)
And Aunt kate is missing from this list. She is the youngest of Grandpa
patrick McManus' siblings. When she was a girl, during the Great
Depression and Dust Bowl days, the story is that her parents couldn't
afford to keep Kate with them in Nebraska, so they sent Aunt Kate to a
relative in California. Kate had a son named Dan. No one seems to know
the scoop there. Kate did end up marrying Karl Edwards who was a hunting
guide and worked the oil fields in Wyoming. My father-in-law, Jim, his
father Patrick and Steve used to go up to Wyoming to hunt with Dan in the
1960s and early 1970s. Karl & Kate had 2 adopted boys of Indian decent,
Shando & Wando (spelling?) - one of those boys died. Kate is still alive
today, she lives in Lafayette, Colorado. Dan Edwards died about 10 years
ago, he was only in his 50s.)

In the 1884 List I found
Hugh McManus, Fish Creek
Hugh McManus, Center
J. McManus, Center
Steve's dad James E. was born in Fish Creek. Is there any connection
between Hugh and Patrick McManus? Steve's grandparents Patrick & Elaine
moved to Denver during the late 30s. They both milked at a local dairy
and Patrick worked for Gates Rubber Co. When he was about 5 or 6 years
old, Jim went back to Nebraska and lived with relatives. According to
Steve it was because of a combination of the polio epidemic in Denver and
the fact that the rural school was going to be closed down because there
weren't enough students. Steve isn't sure whether his father went back to
Fish Creek or if he lived with Uncle Bud & Maybelle.

Thanks for being involved with this project. As families move away,
valuable history and great stories are lost.

Debbie Jacobsen-McManus
7007 E. Bayou Gulch Rd.
Parker, CO 80134
303.387.4083