  We define and compute higher rank analogs of Pandharipande-Thomas stable pair
invariants in primitive classes for K3 surfaces. Higher rank stable pair
invariants for Calabi-Yau threefolds have been defined by Sheshmani
\cite{shesh1,shesh2} using moduli of pairs of the form $\O^n\into \F$ for $\F$
purely one-dimensional and computed via wall-crossing techniques. These
invariants may be thought of as virtually counting embedded curves decorated
with a $(n-1)$-dimensional linear system. We treat invariants counting pairs
$\O^n\into \E$ on a $\K3$ surface for $\E$ an arbitrary stable sheaf of a fixed
numerical type ("coherent systems" in the language of \cite{KY}) whose first
Chern class is primitive, and fully compute them geometrically. The ordinary
stable pair theory of $\K3$ surfaces is treated by \cite{MPT}; there they prove
the KKV conjecture in primitive classes by showing the resulting partition
functions are governed by quasimodular forms. We prove a "higher" KKV
conjecture by showing that our higher rank partition functions are modular
forms.
