  In the context of linear $f(\mathcal{R},T)=\mathcal{R}+\chi T$ gravity, where
$\mathcal{R}$ is the Ricci scalar, $T$ is the trace of the energy-momentum
tensor, and $\chi$ is a dimensionless parameter, we have obtained exact
analytical and numerical solutions for isotropic perfect-fluid spheres in
hydrostatic equilibrium. Our solutions correspond to two-parametric extensions
of the Tolman III (T-III) and Tolman VII (T-VII) models, in terms of the
compactness $\beta$ and $\chi$. By requiring configurations that exhibit
monotonically decreasing radial profiles for both the energy density and
pressure, compliance with the energy conditions, as well as subluminal speed of
sound, we have constrained the parametric space of our solutions. We have also
obtained analytically a parametric deformation of the T-VII solution that
continuously interpolates between the T-III and T-VII models for any $\chi$,
and in the appropriate limits, provides an analytic approximation for the
uniform density configuration in linear $f(\mathcal{R},T)$ gravity. Finally, by
integrating numerically the TOV equations, we have obtained a numerical
solution for the uniform-density configuration and subsequently, using the
mass-radius relations, we have obtained the maximum mass that can be supported
by such configurations. We have found that in the appropriate parametric regime
our solution is in very good agreement with the observational bounds for the
masses and radii of neutron stars.
