What is an example of a $n$ dimensional manifold $M$ which is not a lie group or $S^{7}$ but satisfies the following property?:

There is an $n$ dimensional sub vector space $V\subset \chi^{\infty}(M)$ such that every $0 \neq X \in V$ is a nonvanishing vector field on $M$.

What is an example of a $n$ dimensional manifold $M$ which is not a lie group or $S^{7}$ but satisfies the following property?:

There is an $n$ dimensional sub vector space $V\subset \chi^{\infty}(M)$ such that every $0 \neq X \in V$ is a nonvanishing vector field on $M$.

So this is a motivation to define an invariant of manifolds as follows: $$ \text{The maximum number $k\leq n$ with a k dimensional subvector space$$ $V\subset\chi^{\infty}(M)$ with the above property} $$