Welcome, Waltraud!
It will take some crawling around with pruning shears, bottle, and paintbrush. The shoots need to be cut off at ground level, then IMMEDIATELY painted with napthalene acetic acid (sometimes sold as a root stimulator). It's a case of cut...paint...cut...paint. If there is more than 3 minutes delay between cutting and painting, the hormone will not be absorbed, and the shoot will grow back. This shouldn't harm any of the other plants, or the parent lilac. As a preventative, avoid damaging the roots of the lilac: everywhere a root is cut or bruised, a shoot will grow. So mulch rather than cultivate, keep foot traffic to a minimum (creative stepping stones), and wage limited war on gophers, moles, and voles (Castor bean extract) and grubs (beneficial nematodes or milky spore). Good luck!

many thanks for your advice anwser to my problem , waltraud