  Let $k$ be an algebraically closed field of any characteristic. Let $X$ be a
polarized irreducible smooth projective algebraic variety over $k$. We give
criterion for semistability and stability of system of Hodge bundles on $X$. We
define notion of generalized opers on $X$, and prove semistability of the Higgs
bundle associated to generalized opers. We also show that existence of partial
oper structure on a vector bundle $E$ together with a connection $\nabla$ over
$X$ implies semistability of the pair $(E, \nabla)$.
