What is the name of this variation: 1. e4 b6 2. e5? I had started playing Owen's Defense. My opponent pushed the pawn one step forward.

What variation did he try, securing this pawn he may play well.

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I don't think that this variation has a name. Variations that are infrequently played are less likely to be named. I don't have access to Megabase right now, but I can't even find a game starting with 1. e4 b6 2. e5 in online databases. You're already well away from main lines with 1... b6. 2. e5 seems weird (2. d4 would be my choice) and 2... Nc6 will impede the bishop on b7.