Incomplete normal modal logic systems Apart from the classical example of KH, given by axiom $\Box(\Box p\leftrightarrow p)\to \Box p$, are there any other examples of incomplete propositional normal modal logic systems defined by axioms in 1 variable and of modal degree $\le 2$? If so, I would also appreciate a reference to where the proofs can be found. I am looking for the simplest possible examples. Van Benthem's is of modal degree 3, others are in 2 variables etc.

The answer can be found in the Volume 3 of the _Handbook of Philosophical Logic, 2nd Edition_ , 2001. See the "Correspondence Theory" chapter by Johan van Benthem, pp. 325-408. On page 333, Fact 5, he gives an example of an incomplete modal logic:

The logic $L$ that extends $\mathbf{K}$ with axioms (of degree at most 2 and with only 1 variable) $$ \begin{array}{l} \Box p\to p\\\ \Box\Diamond p\to\Diamond\Box p\\\ \Box(p\to\Box p)\to(\Diamond p\to p) \end{array} $$ is even first-order definable, as it defines the class of frames satisfying the first-order sentence $\forall x\forall y(xRy\leftrightarrow x=y)$, but the modal formula ${\Box p\leftrightarrow p}$, which is valid on this class of frames, is not derivable in $L$. Therefore, the logic $L$ is not Kripke complete.