Non-linear Lie conformal algebras with three generators

We classify certain non-linear Lie conformal algebras with three generators, which can be viewed as deformations of the current Lie conformal algebra of sl(2). In doing so we discover an interesting 1-parameter family of non-linear Lie conformal algebras R(-1)(d) (d is an element of N) and the corresponding freely generated vertex algebras V d 1, which includes for d = 1 the affine vertex algebra of sl(2) at the critical level k = -2. We construct free-field realizations of the algebras V(-1)(d) extending the Wakimoto realization of (sl(2)) over cap at the critical level, and we compute their Zhu algebras.