Star Alliance members Lufthansa, Swiss and United have an airfare deal around $580 round trip for flights from Los Angeles to Cape Town, South Africa for dates in March 2020. This appears to be a developing flight deal as airline routing options and dates have changed and expanded as I wrote this piece. Google Flights shows low fare departure dates on March 7-9 and March 16 at $577-$615 with United Airlines, Lufthansa and Swiss ticketing options. Ticket prices are also low in October and November. I see return dates as short as one week or even for trips longer than a month.
Luthansa ticket on Swiss Airlines Los Angeles LAX – Zurich ZRH – Cape Town CPT – Zurich (18 hr layover) – LAX