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Let $X$ be a L\\'{e}vy process on $\\mathbf{R}^n$ with L\\'{e}vy-Khintchine exponent $(a,A,\\mu)$. {First, we show that if $A$ is non-degenerate then $X$ satisfies (H). Second, under the assumption that $\\mu({\\mathbf{R}^n\\backslash \\sqrt{A}\\mathbf{R}^n})<\\infty$, we show that $X$ satisfies (H) if and only if the equation $$ \\sqrt{A}y=-a-\\int_{\\{x\\in {\\mathbf{R}^n\\backslash \\sqrt{A}\\mathbf{R}^n}:\\,|x|<1\\}}x\\mu(dx),\\ y\\in \\mathbf{R}^n, $$ has at least one solution. 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