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The paper shows the integrability of certain classes of ninth-order dynamic systems that are homogeneous in part, in which the system on the tangent bundle to four-dimensional manifolds is distinguished. In this case, the force fields have various dissipation and generalize the previously considered ones.

The paper shows the integrability of certain classes of seventh-order dynamic systems that are homogeneous in part, in which the system on the tangent bundle to three-dimensional manifolds is distinguished. In this case, the force fields have various dissipation and generalize the previously considered ones.

The unique features of nanomaterials make them quite attractive for applications [1, 2]. One of the factors that is responsible for the anomalous behavior of nanomaterials is due to the surface effects. In particular, taking into account the surface tension allows us to describe the observed finite size effect, i.e., the dependence of the properties of the material, e.g., its Young modulus, on the sizes of the specimen. Experimental and theoretical studies of the influence of the surface tension on effective properties of nanomaterials (nanoporous media, nanofiners, etc.) were described in a significant number of papers (see, e.g., [3-6]). In the present work, we consider the problem of bending of a nanosize plate, taking into account the action of the surface effects. We obtain expressions for the effective stiffnesses.