Time invariant system with integral

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I have this signal $ y(t)= \int_{\infty}^{\infty} x(\tau ) z(t -\tau ) d\tau $ and I have to find if this is time invariant or not. I have some problems with the change of variable. I don’t know if I have to change the variable also into z argument. I made it but my result is that this is time variant and the correct result is the opposite. Thank you so much