If you have a time-constant covariate $X$, you could include it but drop the individual fixed effects $\sigma_i$. (There would be perfect multicollinearity if you kept them.) Since you mentioned that $X$ might have a moderating effect w.r.t. $Z$, you would include interaction terms between $X$ and $Z$. So you would have something like $$ O_{i,t} = \alpha + \beta_t + \gamma x_i + \delta Z_{i,t} + \theta (x_i\cdot Z_{i,t}) + \epsilon_{i,t} $$ with $\beta_t$ being time fixed effects. You could set $\beta_1=0$ for identification. (One more linear restriction might be needed.) From your verbal description of the problem, I am not sure what $\sum_{\substack{k=S \\ k\neq -1}}^{F}{\mu_k}$ is meant to represent, so I did not include it in the equation above.

_{(I might be completely wrong. It has been a long while since I did anything with panel data models.)}

If you have a time-constant covariate $X$, you could include it but drop the individual fixed effects $\sigma_i$. (There could be perfect multicollinearity if you kept them.) Since you mentioned that $X$ might have a moderating effect w.r.t. $Z$, you would include interaction terms between $X$ and $Z$. So you would have something like $$ O_{i,t} = \alpha + \beta_t + \gamma x_i + \delta Z_{i,t} + \theta (x_i\cdot Z_{i,t}) + \epsilon_{i,t} $$ with $\beta_t$ being time fixed effects. You could set $\beta_1=0$ for identification. (One more linear restriction might be needed.) From your verbal description of the problem, I am not sure what $\sum_{\substack{k=S \\ k\neq -1}}^{F}{\mu_k}$ is meant to represent, so I did not include it in the equation above.

_{(I might be completely wrong. It has been a long while since I did anything with panel data models.)}