Let \(\alpha_ 1,\dots,\alpha_ n\) be given real numbers such that \(\alpha_ j\geq 1\) for all \(1\leq j\leq n\), and let \(\delta_ t=\exp(A\ln t)\) (\(t>0\)), where \(A=\text{diag}\{\alpha_ 1,\dots,\alpha_ n\}\). Let \(\rho(x)\) be the unique positive number satisfying \(\delta_{\rho(x)^{-1}}x\in\Sigma^{n-1}=\bigl\{x:\;\sum_{i=1}^ n \alpha_ i^{-1} x_ i^ 2=1\bigr\}\). A nonisotropic singular integral operator \(T_ 0\) is defined by \[ T_ 0 f(x)=\lim_{\varepsilon\to 0} \int_{\rho(y)\geq\varepsilon} h(\rho(y))\Omega(\delta_{\rho(y)^{-1}}y)\rho(y)^{-\alpha}f(x- y)dy, \] where \(n\geq 2\), \(h\in L^ \infty(0,\infty)\), \(\alpha=\sum_{i=1}^ n \alpha_ i\) and \(\int_{\Sigma^{n-1}} \Omega(x)d\sigma(x)=0\). The author establishes the following theorems. Theorem 1. If \(\Omega\in L\log^ + L(\Sigma^{n-1})\), then \(T_ 0\) is bounded on \(L^ 2(\mathbb{R}^ n)\). Theorem 2. If \(\Omega\in L^ q(\Sigma^{n-1})\) for some \(1