Is there an analytical solution to MLE of linear regression with non-normal(exponential) error term?

Question

I know that MLE of normal linear regression $y = k_1x_1+k_2x_2 + \epsilon, \epsilon\sim N(0,1)$ has a nice analytical solution. But what if

1. the error term is exponential distribution?
2. the error term is a distribution in exponential family and the mean linearly depends on $x_i$?