English Translation
I ran to the station not to miss the train.
Considering what is said on correre - Wiktionary, “auxiliary (action towards a goal) essere or (action considered in itself, or transitive) avere”, I wonder why this isn’t “sono corso”.
I somehow think this is different to “ho corso dieci chilometri stamattina”, as this has a clear target. What do you think?
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According to WordReference,
This intransitive verb can take either avere or essere in compound tenses, with little change in meaning.
Not sure that helps us much though!
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I suppose one way to work around the entire issue is to use the passato remoto: Corsi fino alla stazione. 
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I subscribe to this chap on YouTube and here is his take on the matter (he does get there eventually!)
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As a native Italian speaker, I must admit I was not aware of formal rules on this.
But “ho corso fino alla stazione” seems pretty natural in this case.
“Sono corso” also works, but I would omit “fino” in this case.
“Ho corso alla stazione” definitely would be an error.
I think the difference between “ho corso fino alla stazione” and “sono corso alla stazione” is that in the first case, I am saying that I went to the station, all the time running, while in the second form, I am saying simply that I ran (hurried) to the station.
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This explanation is very helpful. Mille grazie @mike-lima
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