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She was blamed for having lied about the case.
Literal
She [topic-は] that case [about-について] lie [object-を] told [quotative-と] saying blamed-was.
A compact accusation sentence. 嘘をつく ('to tell a lie,' lit. 'attach a lie') is a fixed verb-noun pair — つく here is the verb 吐く / 付く meaning 'to attach, emit.' ~といって is a literary reason connector: と (quotative) + 言って ('saying, citing'), giving the reason for what follows: 'on the grounds of saying [X].' 責められる is the passive of 責める ('to blame, criticize'). Note the structure: the lying is the cited reason for blame, not the substance of the blame itself.