Invariants of the extended twisted $h$-Yangian

We investigate the extended twisted $h$-Yangian ${\rm Y}_{N,h}^{\text{tw}}$, a certain algebra which admits both the orthogonal and the symplectic $h$-Yangian as its quotients. We show that ${\rm Y}_{N,h}^{\text{tw}}$ is naturally equipped with the structure of restricted module for a certain algebra ${\rm DY}_{N,h,c}^{tw}$, which resembles the quantum double, as well as with the structure of $\phi$-coordinated quasi module for the Etingof-Kazhdan quantum affine vertex algebra associated with the trigonometric $R$-matrix of type $A$. Finally, we demonstrate how the elements of the quantum Feigin--Frenkel center give rise to explicit formulas for central elements of ${\rm DY}_{N,h,c}^{tw}$ and invariants of ${\rm Y}_{N,h}^{\text{tw}}$ at the critical level, as well as to commutative families in the orthogonal and symplectic $h$-Yangians.