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Two new applications of a technique for spaceability are given in this paper.For the first time this technique is used in the investigation of the algebraicgenericity property of the weak form of Peano's theorem on the existence ofsolutions of the ODE $u'=f(u)$ on $c_0$. The space of all continuous vectorfields $f$ on $c_0$ is proved to contain a closed $\mathfrak{c}$-dimensionalsubspace formed by fields $f$ for which -- except for the null field -- theweak form of Peano's theorem fails to be true. The second applicationgeneralizes known results on the existence of closed $\mathfrak{c}$-dimensionalsubspaces inside certain subsets of $\ell_p(X)$-spaces, $0 < p < \infty$, tothe existence of closed subspaces of maximal dimension inside such subsets.

Lineability and spaceability for the weak form of Peano's theorem and vector-valued sequence spaces