<p>Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(K= \textbf{Q}(\sqrt{d})\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>K</mi> <mo>=</mo> <mi mathvariant="bold">Q</mi> <mo stretchy="false">(</mo> <msqrt> <mi>d</mi> </msqrt> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> a quadratic field and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(\mathcal {O}_{K}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation> its ring of integers. We study the solvability of the Diophantine equation <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(r + s + t = rst = 2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>+</mo> <mi>s</mi> <mo>+</mo> <mi>t</mi> <mo>=</mo> <mi>r</mi> <mi>s</mi> <mi>t</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation> in <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathcal {O}_{K}\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi mathvariant="script">O</mi> <mi>K</mi> </msub> </math></EquationSource> </InlineEquation>. We prove that except for <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(d= -7, -1, 17\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>d</mi> <mo>=</mo> <mo>-</mo> <mn>7</mn> <mo>,</mo> <mo>-</mo> <mn>1</mn> <mo>,</mo> <mn>17</mn> </mrow> </math></EquationSource> </InlineEquation>, and 101 this system is not solvable in the ring of integers of other quadratic fields.</p>