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The aim of this paper is to characterize a Riemannian 3-manifold <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>M</mi><mn>3</mn></msup></semantics></math></inline-formula> equipped with a semi-symmetric metric <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ξ</mi></semantics></math></inline-formula>-connection <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mo>∇</mo><mo>˜</mo></mover></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-Einstein and gradient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-Einstein solitons. The existence of a gradient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ρ</mi></semantics></math></inline-formula>-Einstein soliton in an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>M</mi><mn>3</mn></msup></semantics></math></inline-formula> admitting <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mover accent="true"><mo>∇</mo><mo>˜</mo></mover></semantics></math></inline-formula> is ensured by constructing a non-trivial example, and hence some of our results are verified. By using standard tensorial technique, we prove that the scalar curvature of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><msup><mi>M</mi><mn>3</mn></msup><mo>,</mo><mover accent="true"><mo stretchy="false">∇</mo><mo>˜</mo></mover><mo>)</mo></mrow></semantics></math></inline-formula> satisfies the Poisson equation <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>Δ</mo><mi>R</mi><mo>=</mo><mfrac><mrow><mn>4</mn><mo>(</mo><mn>2</mn><mo>−</mo><mi>σ</mi><mo>−</mo><mn>6</mn><mi>ρ</mi><mo>)</mo></mrow><mi>ρ</mi></mfrac></mrow></semantics></math></inline-formula>.