One-side liquidity provision in automated market-makers

We propose a method for one-side liquidity provision in automated market-making systems with accumulated fees as additional reserves through an atomic exchange of the required fraction of the sending assets followed by a consistent (balanced) investment. For Uniswap invariant an exact solution is obtained, for Balancer and Curve results of numeric simulations are provided. Next, an asymptotic expansion to the second order is proposed, which may be useful for practical investigation of such primitive. Additionally, the Newton -Raphson iterative procedure converging to the exact solution is described and the influence of the accuracy order for the initial approximation is assessed. Finally, conclusions and tips for practical implementation of the proposed primitive for one-side liquidity provision as a smart-contract of modern blockchain networks are given.