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We reconsider the problem of the orbital dynamics of the innermost exoplanetof the $\upsilon$-Andromedae system (i.e., $\upsilon$-And $\mathbf{b}$) intothe framework of a Secular Quasi-Periodic Restricted Hamiltonian model. Thismeans that we preassign the orbits of the planets that are expected to be thebiggest ones in that extrasolar system (namely, $\upsilon$-And $\mathbf{c}$ and$\upsilon$-And $\mathbf{d}$). The Fourier decompositions of their secularmotions are injected in the equations describing the orbital dynamics of$\upsilon$-And $\mathbf{b}$ under the gravitational effects exerted by thosetwo exoplanets. By a computer-assisted procedure, we prove the existence of KAMtori corresponding to orbital motions that we consider to be very robustconfigurations, according to the analysis and the numerical explorations madein our previous article. The computer-assisted assisted proofs are successfullyperformed for two variants of the Secular Quasi-Periodic Restricted Hamiltonianmodel, which differs for what concerns the effects of the relativisticcorrections on the orbital motion of $\upsilon$-And $\mathbf{b}$, depending onwhether they are considered or not.

Computer-assisted proofs of existence of KAM tori in planetary dynamical models of $\upsilon$-And $\mathbf{b}$