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The author considers the Nijenhuis map assigning to two type (1,1) tensor fields $\alpha$, $\beta$ a mapping $$\langle \alpha ,\beta \rangle :(\xi ,\zeta )\mapsto \left[\alpha \right(\xi ),\beta (\zeta \left)\right]+\alpha \circ \beta \left(\right[\xi ,\zeta \left]\right)-\alpha \left(\right[\xi ,\beta \left(\zeta \right)\left]\right)-\beta \left(\right[\alpha \left(\xi \right),\zeta \left)\right]),$$ where $\xi$, $\zeta$ are vector fields. Then $\langle \alpha ,\beta \rangle$ is a type (2,1) tensor field (Nijenhuis tensor) if and only if $[\alpha ,\beta ]=0$. Considering a smooth manifold $X$ with a smooth action of a Lie group, a secondary invariant may be defined as a mapping whose area of invariance is restricted to the inverse image of an invariant subset of $X$ under another invariant mapping. The author recognizes a secondary invariant related to the...