Yes, there are similar modulos to other powers. There are no integer solutions to $$x^5+y^5+z^5+t^5=5$$, as this can be proved modulo $11$.
Also, there only integer solution to $$x^2+3y^2=2z^2$$ is the trivial $(x,y,z)=(0,0,0)$, as this can be proved by applying the $p$-adic method for $p=2$.