Hello,
Thanks  for  the  query to H.C.M. Forum.
Non productive cough ( 
dry cough),
Non responding to antibiotics,
Breathlessness,
      burp after attack of cough( as body want to relieve pressure from stomach).
      When cough stops ( as in your case) , there is burp from stomach , so that extra gas from stomach come out.
All these above mentioned cough related symptoms were related to Allergic 
bronchitis .
There is 
wheezing ( whistling sound from lung during inhalation or exhalation).
 Causes of this type ( dry hacking cough may be),
Mite,
Dust,
Pollen,
spray,
fumes,
smokes,
Fine powder,
hay.
Physical examination by a physician will help you very much. 
X ray, usually did not shows any thing wrong. 
Hope I have answered  your query. 
If further any question I will be happy to help. 
Good luck 
Dr. HET