  Networks of limit cycle oscillators can show intricate patterns of
synchronization such as splay states and cluster synchronization. Here we
analyze dynamical states that display a continuum of seemingly independent
splay clusters. Each splay cluster is a block splay state consisting of
sub-clusters of fully synchronized nodes with uniform amplitudes. Phases of
nodes within a splay cluster are equally spaced, but nodes in different splay
clusters have an arbitrary phase difference that can be fixed or evolve
linearly in time. Such coexisting splay clusters form a decoupled state in that
the dynamical equations become effectively decoupled between oscillators that
can be physically coupled. We provide the conditions that allow the existence
of particular decoupled states by using the eigendecomposition of the coupling
matrix. Additionally, we provide an algorithm to search for admissible
decoupled states using the external equitable partition and orbital partition
considerations combined with symmetry groupoid formalism. Unlike previous
studies, our approach is applicable when existence does not follow from
symmetries alone and also illustrates the differences between adjacency and
Laplacian coupling. We show that the decoupled state can be linearly stable for
a substantial range of parameters using a simple eight-node cube network and
its modifications as an example. We also demonstrate how the linear stability
analysis of decoupled states can be simplified by taking into account the
symmetries of the Jacobian matrix. Some network structures can support multiple
decoupled patterns. To illustrate that, we show the variety of qualitatively
different decoupled states that can arise on two-dimensional square and
hexagonal lattices.
