  The Tolman VII solution is considered by some as one of the few analytical
solutions to Einstein's equations, which describes approximately well the
interior of neutron stars (NSs). This solution is characterized by the mass
$M$, radius $R$, and an energy density that varies quadratically with the
radial coordinate $r$. Recently, Jiang and Yagi proposed a modification of this
solution, the so-called modified Tolman VII (MTVII) solution, by introducing an
additional quartic term to the energy density radial profile. The MTVII
solution is an approximate solution to Einstein's equation, which includes a
new parameter $\alpha$ that allows the solution to have a better agreement with
the energy density profiles for realistic NSs. Here we consider the MTVII
solution, showing that for certain values of the parameter $\alpha$ and
compactness $\mathcal{C}$ this solution manifests a region of negative pressure
near the surface which leads to negative values of the tidal Love number. To
alleviate these drawbacks, we introduce an exact version of the MTVII solution
obtained by solving numerically Einstein's equations for the MTVII energy
density profile. As an application of our new exact MTVII (EMTVII) solution, we
calculate the tidal Love number and tidal deformability, as a function of
$\mathcal{C}$, for different values of the parameter $\alpha$. We find that the
EMTVII solution predicts a positive tidal Love number for the whole range of
allowed values of parameters $(\mathcal{C},\alpha)$, in agreement with previous
results for realistic NSs.
