Non-homogeneous Beatty sequences leading to invariant games

We characterize pairs of complementary non-homogeneous Beatty sequences $(A_n)_{n>0}$ and $(B_n)_{n>0}$, with the restriction $A_1=1$ and $B_1\geq 3$, for which there exists an invariant take-away game having $\{(A_n,B_n),(B_n,A_n)\mid n> 0\}\cup\{(0,0)\}$ as set of $P$-positions. Using the notion of Sturmian word arising in combinatorics on words, this characterization can be translated into a decision procedure relying only on a few algebraic tests about algebraicity or rational independence. This work partially answers to a question of Larsson et al. raised in Larsson et al.