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We construct novel solutions to the set-theoretical entwining Yang-Baxter
equation. These solutions are birational maps involving non-commutative
dynamical variables which are elements of the Grassmann algebra of order $n$.
The maps arise from refactorisation problems of Lax supermatrices associated to
a nonlinear Schr\"odinger equation. In this non-commutative setting, we
construct a spectral curve associated to each of the obtained maps using the
characteristic function of its monodromy supermatrix. We find generating
functions of invariants (first integrals) for the entwining Yang-Baxter maps
from the moduli of the spectral curves. Moreover, we show that a hierarchy of
birational entwining Yang-Baxter maps with commutative variables can be
obtained by fixing the order $n$ of the Grassmann algebra. We present the
members of the hierarchy in the case $n=1$ (dual numbers) and $n=2$, and
discuss their dynamical and integrability properties, such as Lax matrices,
invariants, and measure preservation.
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