Fix an algebraically closed field $k$, an algebraic one-dimensional torus $G_m$ and a non-singular scheme $X$ of finite type over $k.$

Let us define the following:

**Condition 1:** $X$ can be covered by $G_m$-invariant quasi-affine open subschemes.

In the paper "Some theorems on actions of algebraic groups" (The Annals of Mathematics, Second Series, Vol. 98, No. 3 (Nov., 1973), pp. 480-497), Bialynicki-Birula constructs, roughly speaking, to any action of $G_m$ on $X$, satisfying Condition 1, two canonical decompositions of $X$ into non-singular $G_m$-invariant locally closed subschemes (Theorem 4.1).

Moreover, Bialynicki-Birula states that if $X$ is projective, Condition 1 is automatically satisfied (he cites Kambayashi, Projective representations of algebraic groups of transformations, Amer. J. Math. 88 (1966), 199-205.)

**Question:** Assume that $X$ is a non-singular quasi-projective scheme over $k.$ Under what extra assumptions, does $X$ satisfy Condition 1?