We use Frolicher-Nijenhuis theory to reformulate the four Helmholtz conditions[3,4] for the inverse problem of the calculus of variations in terms of semi-basic 1-forms and linear partial differential operators, three of order one and one of order two[1].In the homogeneous case, which corresponds to projective metrizability and Finsler metrizability, we prove that some of the Helmholtz conditions dependent on the other ones.More exactly, we show that a spray is projectively metrizable if and only if two Helmholtz conditions are satisfied and it is Finsler metrizable if and only if three Helmholtz conditions are satisfied.We use Spencers technique [2] to address the formal integrabillity of partial differential operators that define the Helmholtz conditions.A special attention is paid to the formal integrability of these operators in the homogeneous case.