We consider an $f(3,\varepsilon)$-structure manifold $\mathcal{M}^n$ having $L$ and $M$ as complementary distributions and study its invariant submanifolds. Two cases of invariant submanifolds are considered. In the first case we get an induced almost complex structure or almost product structure, and in the second case we obtain an induced $\tilde f(3,\varepsilon)$-structure on the invariant submanifold. It is shown that the complementary distribution of induced $\tilde f(3,\varepsilon)$-structure in the submanifold embedded as an invariant submanifold of $\mathcal{M}^n$ and the induced $\tilde f(3,\varepsilon)$-structure are all integrable if the corresponding distributions and the structure are integrable in $\mathcal{M}^n$.