| Summary: | The subject of the present paper is the investigation of a new type of solitons, called <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-∗-Ricci solitons in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow></semantics></math></inline-formula>-almost co-Kähler manifold (briefly, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>c</mi><mi>k</mi><mi>m</mi></mrow></semantics></math></inline-formula>), which generalizes the notion of the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-Ricci soliton introduced by Cho and Kimura. First, the expression of the ∗-Ricci tensor on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>c</mi><mi>k</mi><mi>m</mi></mrow></semantics></math></inline-formula> is obtained. Additionally, we classify the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-∗-Ricci solitons in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>c</mi><mi>k</mi><mi>m</mi></mrow></semantics></math></inline-formula>s. Next, we investigate <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>(</mo><mi>k</mi><mo>,</mo><mi>μ</mi><mo>)</mo></mrow></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>a</mi><mi>c</mi><mi>k</mi><mi>m</mi></mrow></semantics></math></inline-formula>s admitting gradient <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>η</mi></semantics></math></inline-formula>-∗-Ricci solitons. Finally, we construct two examples to illustrate our results.
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