| Summary: | A single master equation governs the behaviour of shear-free neutral perfect fluid distributions arising in gravity theories. In this paper, we study the integrability of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>y</mi><mrow><mi>x</mi><mi>x</mi></mrow></msub><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><msup><mi>y</mi><mn>2</mn></msup><mo>,</mo></mrow></semantics></math></inline-formula> find new solutions, and generate a new first integral. The first integral is subject to an integrability condition which is an integral equation which restricts the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>x</mi><mo>)</mo><mo>.</mo></mrow></semantics></math></inline-formula> We find that the integrability condition can be written as a third order differential equation whose solution can be expressed in terms of elementary functions and elliptic integrals. The solution of the integrability condition is generally given parametrically. A particular form of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>∼</mo><mfrac><mn>1</mn><msup><mi>x</mi><mn>5</mn></msup></mfrac><msup><mfenced separators="" open="(" close=")"><mn>1</mn><mo>−</mo><mfrac><mn>1</mn><mi>x</mi></mfrac></mfenced><mrow><mo>−</mo><mn>15</mn><mo>/</mo><mn>7</mn></mrow></msup></mrow></semantics></math></inline-formula> which corresponds to repeated roots of a cubic equation is given explicitly, which is a new result. Our investigation demonstrates that complexity of a self-gravitating shear-free fluid is related to the existence of a first integral, and this may be extendable to general matter distributions.
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