Based on the dAlembert-Lagrange-Poincar{e} variational principle, we formulate general equations of motion for mechanical systems subject to nonlinear nonholonomic constraints, that do not involve Lagrangian undetermined multipliers. We write these equations in a canonical form called the Poincar{e}-Hamilton equations, and study a version of corresponding Poincar{e}-Cartan integral invariant which are derived by means of a type of asynchronous variation of the Poincar{e} variables of the problem that involve the variation of the time. As a consequence, it is shown that the invariance of a certain line integral under the motion of a mechanical system of the type considered characterizes the Poincar{e}-Hamilton equations as underlying equations of the motion. As a special case, an invariant analogous to Poincar{e} linear integral invariant is obtained.
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