A remark on independence of a tubular statistic and the sample mean

Given a sample of size $n$ from a distribution with density $y(x)$, we show that if a definite $n-1$-dimensional tubular statistic (i.e. a continuous function on $R^n$ reduceable by an orthogonal transformation to a function on $R^{n-1}$ vanishing only at the origin) and the sample mean are independent then $y(x)$ is normal.