  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.
