Today I will explain the fifth freedom flights for Ethiopian Airlines(ET), an Ethiopian airline that belongs to Star alliance. It is quite complicated route though, let me explain what it is like.　

BLZ – HRE – LUN, LUN – BLZ, HRE – BLZ

HRE <-> BLZ(300 miles)
21SEP – 28SEP
C : JPY155,000 –
HRE -> LUN(250 miles)
21SEP – 28SEP
C : JPY115,000 –
LUN -> BLZ(435 miles)
18SEP – 25SEP
C : 97,400 –
ET34

It is all outside of Ethiopia, and the route changes depending on the date.　

ADD – LFW – GRU

LFW <-> GRU(3811 miles)
ET506/507
19OCT – 27OCT
C : JPY350,770 –

ADD – LFW – EWR

LFW <-> EWR(5181 miles)
ET508/509
17NOV – 25NOV
C : JPY430,850 –

ADD – HKG – ICN

HKG <-> ICN(1284 miles)
ET608/609
20OCT – 27OCT
C : JPY74,650 –

ADD – BKK – KUL

BKK <-> KUL(754 miles)
ET618/619
21OCT – 28OCT
C : JPY29,780 –

I think it is worth checking for because you might find quite a good price.

ADD – HKG – NRT

HKG <-> NRT(1842 miles)
ET672/673
21SEP – 28SEP
C : JPY100,770 –

ADD – ARN – OSL

ARN <-> OSL(240 miles)
ET714/715, ET1692, 1694/ET1693, 1695

It seems you cannot buy only this route.

ADD – VIE – ARN

VIE <-> ARN(800 miles)
ET724/725
21OCT – 28OCT
C : JPY86,240 –

ADD – KGL – BJM

KGL <-> BJM(109 miles)
ET806/807
19OCT – 26OCT
C : JPY52,780 –

LUN – HRE – ADD

LUN -> HRE(250 miles)
ET863
22OCT – 28OCT
C : JPY80,250 –

ADD – HRE – LUN, LUN – ADD

HRE -> LUN(250 miles)
ET873
24OCT – 30OCT
C : JPY : 117,490 –

ADD – LUN – HRE, HRE – ADD

LUN -> HRE(250 miles)
ET893
22SEP – 29SEP
C : JPY58,170 –

Only departures from ADD to HRE will call at LUN.

ADD – BKK – SIN

BKK <-> SIN(875 miles)
ET1306/1307

It seems you cannot buy the fifth freedom flight route only.

NRT – HKG

NRT <-> HKG(1842 miles)
ET1400/1401
21SEP – 28SEP
C : JPY100,770 –

Conclusion

There are lots of routes for LUN – HRE.

Also almost all of them are irregular flights.

Still I think you can keep checking the BKK – KUL route.