How to find an instrumental variable

Let's consider the linear model with endogenous errors: $$y_t = \beta x_t + e_t$$ with $$E[x_t e_t] \neq 0$$.

How do we find in practice an instrumental variable (IV)? i.e. a variable $$z_t$$ that verifies: \begin{align} E[z_t x_t] \neq 0 \\ E[z_t e_t] = 0 \end{align}

In practice, we have measures of $$x$$, $$y$$ and some $$z$$. The measure of $$z$$ is an instrument, i.e., satisfies the conditions you specified, if it affects $$x$$ and affects $$y$$ only through its effect on $$y$$. The only way to find such a valid measure $$z$$ is to understand the context one is studying; in particular, one needs a good, contextual understanding of the relationship between $$x$$ and $$y$$, and why $$z$$ would affect $$y$$ only through its effect on $$x$$. There is no purely theoretical way to find a good instrument. One has to consider the "true world" outside of mathematics. I suggest you read some applications that use the instrumental variables strategy, for example in epidemiology or economics.