Finding a reaction from a KEGG map
It was a simple search. You could have easily searched for glucose oxidase in KEGG, and you would have got this reaction.

In fact if you had clicked the link for glucunolactone's compound page, you would have got a list of many different reactions that convert glucose to glucunolactone. 

You can even click the arrow which opens the KEGG-ORTHOLOGY page. You can find a field called RN in the section called "Other DBs". The links corresponding to this field would give you the exact reactions, that the pathway uses.