Integrable billiards, Poncelet porisms and Pencils of Quadrics
We study billiard systems within pencils of quadrics in $d$ dimensional space. We consider a set $T$ of lines tangent to the same $d-1$ quadrics from the confocal family. An algebraic structure can be introduced on $T$ and fundamental geometrical property can be derived: any two skew lines from $T$ can be obtained from each other by at most $d-1$ billiard reflections from quadrics of the confocal family. This property gives us a possibility to generalize several classical low dimensional and genus one results.