It is known that (i) a subspace ${\mathcal N}$ of the Hardy space $H^2$ (equal to the reproducing kernel Hilbert space with Szeg\"o reproducing kernel$k_{\rm Sz}(z, \zeta) = 1/(1 -z \overline{\zeta})$) which is invariant under the backward shift operator $(M_z)^*$ (where $M_z \colon f(z)\mapsto z f(z)$ is the forward shift) can be represented as the range of the observability operator of a conservative discrete-time linear system, (ii) the transfer-function of this conservative linear system in turn is the Beurling-Lax representer for the forward-shift invariant subspace ${\mathcal M} : = {\mathcal N}^\perp$, and (iii) this transfer function also serves as the Sz.-Nagy-Foias characteristic function of the pure contraction operator $T$ given by $T = P_{\mathcal N} M_z |_{\mathcal N}$. These three themes have been generalized over the last couple of decades to the more general free noncommutative setting where the Hardy space is replaced by the full Fock space (formal power series in $d$ freely noncommutative indeterminates with absolutely square-summable coefficients), where the shift is replaced by the right shift tuple ${\mathbf S} = (S_{R,1}, \dots, S_{R,d})$ (where $S_{R,j} \colon f(z) \mapsto f(z) \cdot z_j$), where the conservative discrete-time linear system becomes a certain type of conservative multidimensional input/state/output linear system with evolution along a rooted tree with each node having $d$ forward branches, where a backward shift-invariant subspace ${\mathcal N}$ is the range of the observability operator for such conservativemultidimensional linear system, where the transfer function of this system is the Beurling-Lax representer (now a partially isometric multiplier between full Fock spaces) for the forward shift-invariant subspace ${\mathcal M} = {\mathcal N}^\perp$, and where this transfer function also serves asthe characteristic function (in the sense of Popescu) for the row contraction obtained as the compression of the forward shift $d$-tuple ${\mathbf S}$ to the subspace ${\mathcal N}$. The plan of the talk is to focus on the next generalization of more recent origin, where the Hardy space becomes a weighted full Fock space (to be thought of as generalized Bergman-space weights), and themes (i), (ii), (iii) still go through in a precise but more complicated form. The talk reports on joint work with Vladimir Bolotnikov of the College of William \& Mary.