And Australia send some thousands.
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So as a POD lets Erhard win a absolut majority in 1965. I think they would´nt send much more then 1-2000 men, mostly combat support (pioneers, field hospitals).
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But I expect not just protests from the left but also from the right. The NPD was on the rise at this time. They would complain that Germans have to die for American interest´. Propably it would get them above 5% in 1969.
The POD should be easy - the conservatives got over 47 % in 1965, and came very close to a majority of seats. So a small shift e.g. from the (also very strong) liberal democrats should do the trick.
The German constitution is another problem - here some handwaving is needed.
I think the Australian experience in Vietnam could be a blueprint for the German involvement. So Erhard is re-elected in 1965 and is able to form a government without the liberal democrats.
Being aware that any larger deployment will not be popular, a token force
of about 100 or so specialists is send to Vietnam in 1966. This is generally supported by his party, but not overwhelmingly so. There is more critic from the social and liberal democrats (what actually was their position to the request in OTL ?), but nothing too serious.
However, a year later more troops are requested, this time combat troops.
Now Erhard faces more resistance, even inside his party. However, since
the deployment so far was without significant losses, he still has enough support. So in 1967 the first German combat troops are sent, among them
draftees.