We have explored exact solutions free from any physical and geometrical singularities, as well as the existence of compact stellar systems throughout linear and Starobinsky-$f(\mathcal{R},\mathcal{T})-$gravity theory. As we generally well-known, the general exact solutions of the altered Einstein Field Equations (EFEs) in the background of this gravity theory are one of the most complex tasks. To acquire simpler solution of the altered field equations, we consider linear and Starobinsky shape of the algebraic function as $f(\mathcal{R},\mathcal{T})=\mathcal{R}+2\chi \mathcal{T}$ and $f(\mathcal{R},\mathcal{T})=\mathcal{R}+\xi {\mathcal{R}}^2+2\chi \mathcal{T}$ (where $\mathcal{R}$ is scalar curvature, $\mathcal{T}$ is the trace of the stress–energy tensor and $\chi \&\xi$ denotes a coupling constants). In this regards, we propose metric potentials $e^{\lambda }$, and obtain the other metric potentials ($e^{\nu }$) under the embedding class one condition. We then compare the cases when $\xi =\chi =0$ [GR], $\xi =0,\chi =0.5$ [$f_L(\mathcal{R},\mathcal{T})$], $\xi =0.5,\chi =0$ [$f_S(\mathcal{R},\mathcal{T})$] and $\xi =\chi =0.5$ [$f_{S+L}(\mathcal{R},\mathcal{T})$]. The obtained solution is well-behaved in all physical and mathematical points of view. Further, we provided a detailed physical acceptability of the solution by exploring main salient characteristics under different physical analysis in the linear $f(\mathcal{R},\mathcal{T})-$gravity. Moreover, the generalizing $M-R$ and $M-I$ curves from our solution are well fitted with observational data of the three compact objects viz., PSR J1614-2230, Vela X-1 and 4U 1820-30. We also found the beautiful results that the equation of state (EoS) is stiffest in $\chi =0$ than $\chi \ne 0$, and the sensitivity in EoS is better in $I-M$ graph than in $M-R$ graph. Finally, we have successfully represented the effects of all the physical requirements in the arena of $f(\mathcal{R},\mathcal{T})-$gravity and we compared them with the standard GR results which can be recovered at $\chi =0$.

我们探索了没有任何物理和几何奇异性的精确解决方案，以及整个线性和Starobinsky-$F（\mathcal{[R}，\mathcal{Ť}）-$引力理论。众所周知，在这种引力理论的背景下，改变后的爱因斯坦场方程（EFE）的一般精确解是最复杂的任务之一。为了获得变场方程的更简单解，我们将代数函数的线性和Starobinsky形状视为$F（\mathcal{[R}，\mathcal{Ť}）=\mathcal{[R}+2\chi \mathcal{Ť}$ 和 $F（\mathcal{[R}，\mathcal{Ť}）=\mathcal{[R}+\xi {\mathcal{[R}}^2+2\chi \mathcal{Ť}$ （哪里 $\mathcal{[R}$ 是标量曲率， $\mathcal{Ť}$ 是应力-能量张量和 $\chi 和\xi$表示耦合常数）。在这方面，我们提出了度量潜力${Ë}^{\lambda }$，并获得其他度量电位（${Ë}^{\nu }$）在嵌入类别一的条件下。然后我们比较情况$\xi =\chi =0$ [GR]， $\xi =0，\chi =0。5$ [$F_{\mathrm{大号}}（\mathcal{[R}，\mathcal{Ť}）$]， $\xi =0。5，\chi =0$ [$F_{\mathrm{小号}}（\mathcal{[R}，\mathcal{Ť}）$]和 $\xi =\chi =0。5$ [$F_{\mathrm{小号}+\mathrm{大号}}（\mathcal{[R}，\mathcal{Ť}）$]。所获得的解决方案在所有物理和数学观点上都表现良好。此外，我们通过探索线性中不同物理分析下的主要显着特征，提供了解决方案的详细物理可接受性$F（\mathcal{[R}，\mathcal{Ť}）-$重力。而且，泛化$\mathrm{中号}-\mathrm{[R}$ 和 $\mathrm{中号}-\mathrm{一世}$我们的解决方案的曲线很好地拟合了三个紧凑物体的观测数据，即PSR J1614-2230，Vela X-1和4U 1820-30。我们还发现，状态方程（EoS）在$\chi =0$ 比 $\chi \ne 0$，并且EoS的灵敏度在 $\mathrm{一世}-\mathrm{中号}$ 图比 $\mathrm{中号}-\mathrm{[R}$图形。最后，我们已经成功地表示了竞技场中所有身体需求的影响$F（\mathcal{[R}，\mathcal{Ť}）-$重力，我们将它们与标准GR结果进行了比较，可以在 $\chi =0$。