I agree with shibha:
If you create a (temp) table with all possible dates
(from min(start) till max(end) ) only the sundays,
you can join the tables like this:

select o.id, d.start_dt, d.end_dt
from org_table o
, date_table d
where d.start_dt >= o.start_dt
and d.end_dt < o.end_dt

Can be some mistakes in it, but it's the idea.