  In this work, we establish a full single-letter characterization of the
rate-distortion region of an instance of the Gray-Wyner model with side
information at the decoders. Specifically, in this model an encoder observes a
pair of memoryless, arbitrarily correlated, sources $(S^n_1,S^n_2)$ and
communicates with two receivers over an error-free rate-limited link of
capacity $R_0$, as well as error-free rate-limited individual links of
capacities $R_1$ to the first receiver and $R_2$ to the second receiver. Both
receivers reproduce the source component $S^n_2$ losslessly; and Receiver $1$
also reproduces the source component $S^n_1$ lossily, to within some prescribed
fidelity level $D_1$. Also, Receiver $1$ and Receiver $2$ are equipped
respectively with memoryless side information sequences $Y^n_1$ and $Y^n_2$.
Important in this setup, the side information sequences are arbitrarily
correlated among them, and with the source pair $(S^n_1,S^n_2)$; and are not
assumed to exhibit any particular ordering. Furthermore, by specializing the
main result to two Heegard-Berger models with successive refinement and
scalable coding, we shed light on the roles of the common and private
descriptions that the encoder should produce and what they should carry
optimally. We develop intuitions by analyzing the developed single-letter
optimal rate-distortion regions of these models, and discuss some insightful
binary examples.
