Introduction. In the investigation of a non-degenerate quadratic form f (x) with coefficients in a local or global field K, one associates two groups with f (x), the orthogonal and metaplectic groups. If K is, e.g., the field R of real numbers, the second group is the unitary group generated by Ut = exp(it -f(x)) and its conjugate V, under the Fourier transformation for all t in R; and it is locally isomorphic to SL2(R). By using a similar definition one can associate with any polynomial f (x) of higher degree a group. Such groups have appeared in connection with the Siegel-Weil formulas for the norm forms of certain simple Jordan algebras. On the other hand Kubota [4], [5] and Yamazaki [8] have associated for K = C and R metaplectic groups with where n need not be 2, in fact n 2 and n = 2k 2, respectively. In both cases they have used certain Bessel transformations instead of the Fourier transformation to get the metaplectic groups. In a recent paper [3] we have shown that the Lie algebra of a hypermetaplectic group over R is infinite dimensional (at least if the Fourier transformation is defined relative to the bilinear form associated with a definite quadratic form). In this paper we have restricted ourselves to the one-variable case and tried to obtain all metaplectic groups over K = R and C. More precisely we have started from an arbitrary polynomial f (x) of one variable x with coefficients in K = R or C, put U, = exp (it .f (x)) or exp(2iRe(tf(x)) and determined all cases where U, and another oneparameter subgroup V, of the unitary group of a certain L2-space generate a Lie group G locally isomorphic to SL2(K). In doing so we have assumed that the tangent vector or vectors of V, at t = 0 is a linear differential operator with polynomial coefficients, a condition satisfied in the known cases. The result is that f (x) simply becomes x, x2 under a suitable normalization and, accordingly, the differential operator becomes i-times