  The essence of the notion of lineability and spaceability is to find linear
structures in somewhat chaotic environments. The existing methods, in general,
use \textit{ad hoc} arguments and few general techniques are known. Motivated
by the search of general methods, in this paper we formally extend recent
results of G.\ Botelho and V.V. F\'{a}varo on invariant sequence spaces to a
more general setting. Our main results show that some subsets of invariant
sequence spaces contain, up to the null vector, a closed infinite-dimensional
subspace.
