  SPIRou is a near-infrared spectropolarimeter and a high-precision
velocimeter. The SPIRou Legacy Survey collected data from February 2019 to June
2022, half of the time devoted to a blind search for exoplanets around nearby
cool stars. The aim of this paper is to present this program and an overview of
its properties, and to revisit the radial velocity (RV) data of two multiplanet
systems, including new visits with SPIRou. From SPIRou data, we can extract
precise RVs using efficient telluric correction and line-by-line measurement
techniques, and we can reconstruct stellar magnetic fields from the collection
of polarized spectra using the Zeeman-Doppler imaging method. The stellar
sample of our blind search in the solar neighborhood, the observing strategy,
the RV noise estimates, chromatic behavior, and current limitations of SPIRou
RV measurements on bright M dwarfs are described. In addition, SPIRou data over
a 2.5-year time span allow us to revisit the known multiplanet systems GJ~876
and GJ~1148. For GJ~876, the new dynamical analysis including the four planets
is consistent with previous models and confirms that this system is deep in the
Laplace resonance and likely chaotic. The large-scale magnetic map of GJ~876
over two consecutive observing seasons is obtained and shows a dominant dipolar
field with a polar strength of 30~G, which defines the magnetic environment in
which the inner planet with a period of 1.94~d is embedded. For GJ~1148, we
refine the known two-planet model.
