We develop a general framework for numerically solving differential equations while preserving invariants. As in standard projection methods, we project an arbitrary base integrator onto an invariant-preserving manifold, however, our method exploits homogeneous symmetries to evaluate the projection exactly and in closed form. This yields explicit invariant-preserving integrators for a broad class of nonlinear systems, as well as pseudo-invariant-preserving schemes capable of preserving multiple invariants to arbitrarily high precision. The resulting methods are high-order and introduce negligible computational overhead relative to the base solver. When incorporated into adaptive solvers such as Dormand-Prince 8(5,3), they provide error-controlled, invariant-preserving, high-order time-stepping schemes. Numerical experiments on double-pendulum and Kepler ODEs as well as semidiscretised KdV and Camassa-Holm PDEs demonstrate substantial improvements in both accuracy and efficiency over standard approaches.

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