Affrirmative, Natterjack /
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If you have the written FAA ATP and take the FAA ATP/LR-JET check ride, you will obtain what I read you want to have.
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Your FAA certificate (licence) will read as follows -
"Airline Transport Pilot
"Airplane Multiengine Land - LR-JET
A small detail... Apply for an FCC R/T Operator certificate while you are in the USA, since your airplane will be US registry, you are better to have a R/T certificate that goes along with it. No test, only costs you $10.oo or so...
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Your certificate will remain valid provided that you meet FAR 61.58 (PIC) or 61.55 (SIC) requirements and hold a current FAA medical certificate.
The LR-JET type rating is valid on Lear 23/24/25, 28/29, 31, 35/36 and 55 types.
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Happy contrails