Integrability conditions for polynomial structures

VANZURA 0. Let / be a tensor field of type (1, 1) defined on a differentiable manifold and satisfying there a polynomial equation with constant coefficients. Under the assumption that the polynomial nj rQ<i n ~1 + + n -if+ n has only simple roots we give two necessary and sufficient conditions for the integrability of the tensor field /. The integrability conditions presented in the paper generalize those known for various structures on manifolds (e.g. almost complex, almost contact, almost product structure etc.). All differentiable structures involved are supposed to be of class C.