Complexity Class NEXP$^\text{NP}$ I have a problem which is in NEXP$^{\text{NP}}$ and can also be solved by an alternating TM using exponential time and just one alternation (starting in an existential state).

Is there anything known about NEXP$^{\text{NP}}$? Is it equal to NEXP or some other class? Are there complete problems other than the generic one (given an NEXP$^{\text{NP}}$ machine and a word, does it accept?).
A natural $\text{NEXP}^{\text{NP}}$-complete problem is deciding a sentence of Presburger arithmetic with an $\exists^*\forall^*\exists^*$-quantifier prefix (as shown here). Further complete problems related to database theory have been studied here.