  We study the model checking problem of an extended $\mathsf{MSO}$ with local
and global cardinality constraints, called
$\mathsf{MSO}^{\mathsf{GL}}_{\mathsf{Lin}}$, introduced recently by Knop,
Kouteck\'{y}, Masa\v{r}\'{i}k, and Toufar [Log. Methods Comput. Sci., 15(4),
2019]. We show that the problem is fixed-parameter tractable parameterized by
vertex integrity, where vertex integrity is a graph parameter standing between
vertex cover number and treedepth. Our result thus narrows the gap between the
fixed-parameter tractability parameterized by vertex cover number and the
W[1]-hardness parameterized by treedepth.
