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<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi mathvariant="script">R</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula>-gravity is a generalization of Einstein’s field equations <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msub><mrow><mi>E</mi><mi>F</mi><mi>E</mi></mrow><mi>s</mi></msub><mo>)</mo></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi mathvariant="script">R</mi><mo>)</mo></mrow></semantics></math></inline-formula>-gravity. In this research article, we demonstrate the virtues of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi mathvariant="script">R</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula>-gravity model with Einstein solitons <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>E</mi><mi>S</mi><mo>)</mo></mrow></semantics></math></inline-formula> and gradient Einstein solitons <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>G</mi><mi>E</mi><mi>S</mi><mo>)</mo></mrow></semantics></math></inline-formula>. We acquire the equation of state of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi mathvariant="script">R</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula>-gravity, provided the matter of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi mathvariant="script">R</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula>-gravity is perfect fluid. In this series, we give a clue to determine pressure and density in radiation and phantom barrier era, respectively. It is proved that if a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi mathvariant="script">R</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula>-gravity filled with perfect fluid admits an Einstein soliton <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>g</mi><mo>,</mo><mi>ρ</mi><mo>,</mo><mi>λ</mi><mo>)</mo></mrow></semantics></math></inline-formula> and the Einstein soliton vector field <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>g</mi><mo>,</mo><mi>ρ</mi><mo>,</mo><mi>λ</mi><mo>)</mo></mrow></semantics></math></inline-formula> is Killing, then the scalar curvature is constant and the Ricci tensor is proportional to the metric tensor. We also establish the Liouville’s equation in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi mathvariant="script">R</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula>-gravity model. Next, we prove that if a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi mathvariant="script">R</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula>-gravity filled with perfect fluid admits a gradient Einstein soliton, then the potential function of gradient Einstein soliton satisfies Poisson equation. We also establish some physical properties of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi mathvariant="script">R</mi><mo>,</mo><mi>T</mi><mo>)</mo></mrow></semantics></math></inline-formula>-gravity model together with gradient Einstein soliton.