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Aims. The project aims to understand better the role of wide brown dwarf companions on planetary systems. Methods. We obtained high-resolution spectra of six bright stars with co-moving wide substellar companions with the SONG, CARMENES, and STELLA high-resolution spectrographs. We used these spectra to derive radial velocities together with a complete set of stellar physical parameters. We then investigated radial velocities signals and discussed the fraction of planets in such systems. We also re-analyzed the ages of our targets, which were used to derive the physical parameters of wide brown dwarf companions. Finally, a compilation of systems with known planets from the literature is considered along with our sample to search for possible peculiarities in their parameter distributions. Results. Based on the derived ages of six observed systems, we re-computed the masses of the wide companions, confirming their substellar nature. We confirmed planets in the HD 3651 and HIP 70849 systems and found a new planetary candidate in the HD 46588 system. In our survey, which is sensitive mostly to Neptune-mass planets at short periods of a few days and Saturn-mass planets at longer periods of hundreds of days, we derived a frequency of planets orbiting stars with wide brown dwarf companions below 70% with the uncertainties included. Comparing the parameter distributions of our sample with single stars, we observe the enhancement of planets with short periods below six days in systems with a wide stellar companion. Finally, planets in systems with wide BD companions follow their own eccentricity distribution with a maximum at $\sim0.65$ and have periods larger than 40 days, masses larger than $0.1\,M_J$, and eccentricities larger than 0.4., 28 pages, 31 figures, accepted for publication in Astronomy & Astrophysics

56 UMa is a wide binary system that contains a chemically peculiar red giant and a faint companion. Due to its surface chemical abundances, the red giant was classified as a barium (Ba) star. This implies that the companion has to be a white dwarf, since Ba stars form when mass is transferred to them from an s-process rich Asymptotic Giant Branch (AGB) star. However, in the case of 56 UMa, the companion might be too massive to be the progeny of an AGB star that efficiently produced s-process elements such as barium. In this Letter, we revisit the orbital parameters of the system and perform a full spectral analysis with the goal of investigating the Ba-star classification of the giant and unravelling the nature of its faint companion. We combined radial-velocity and astrometric data to refine the orbital parameters of the system, including the orbital inclination and the companion mass. Then, we re-determined the stellar parameters of the giant and its chemical abundances using high-resolution HERMES spectra. Finally, we investigated the morphology of the interstellar gas in the vicinity of the system. The faint component in 56 UMa has a mass of $1.31 \pm 0.12$ M$_{\odot}$, which, together with the mixed s+r abundance profile of the red giant, confirms that the giant is not a standard barium star. Additionally, the clear identification of a cavity surrounding 56 UMa could indicate that a supernova explosion occurred about 10 5 years ago in the system, suggesting that the faint companion might be a neutron star. However, finding an evolutionary scenario that explains all the observables is not trivial, so we discuss different possible configurations of the system and their respective merits., Comment: Accepted for publication in A&AL