## 2 Models of Motion
\begin{array}{l}{v_{x}}^{2}+{v_{y}}^{2}+{n_{S}}^{2}\big{(}x^{2}+y^{2} \big{)}+2n_{S}\big{(}x\ v_{y}-y\ v_{x}\big{)}-\frac{2\mu_{E}}{\sqrt{x^{2}+y^{2 }}}-h=0,\\ x\ v_{y}-y\ v_{x}+n_{S}\big{(}x^{2}+y^{2}\big{)}-\sigma=0,\\ A(t)\left(\begin{array}{c}\sigma\big{(}v_{y}+n_{S}x\big{)}-\mu_{E}\frac{\sqrt {x^{2}+y^{2}}}{\sqrt{x^{2}+y^{2}}}\\ -\sigma(v_{x}-n_{S}y)-\mu_{E}\frac{y}{\sqrt{x^{2}+y^{2}}}\end{array}\right)- \left(\begin{array}{c}\Lambda_{\xi}\\ \Lambda_{\eta}\end{array}\right)=\mathbf{0}\end{array} \right\} \tag{6}\]  
The matrix \(A(t)\) is defined as follows  
\[A(t)= \begin{pmatrix}\cos(n_{S}t)&-\sin(n_{S}t)\\ \sin(n_{S}t)&\cos(n_{S}t)\end{pmatrix} \]  
In the above expressions (6), the indices \(\xi\) and \(\eta\) corresponds to the fixed directions of the coordinate system (CS) axes in inertial space. It is interesting to note that for a system (5) there is one more \"atypical\" (within the framework of the R2 BP) first mechanical integral, which is an analogue of the Jacobi integral from the restricted circular three-body problem (RC3 BP)  
\[\left. \begin{array}{l}{v_{x}}^{2}+{v_{y}}^{2}-{n_{S}}^{2}\big{(}x^{2}+y^{2} \big{)}-\frac{2\mu_{E}}{\sqrt{x^{2}+y^{2}}}-\widetilde{h}=0,\\ h=\widetilde{h}+2n_{S}\sigma\end{array} \right\} \tag{7}\]  
Equation (7) determines an interesting relationship between the energy and \(\sigma\) constants in a planar circular problem. We also should note here that, according to the accepted model of the perturbed motion of the SC (Equations (2)-(4)) it would not be difficult to describe the SC motion under the framework of the RC3 BP Earth-Sun-SC. To do this, in Equation (2a) it suffices to let  
\[\mathbf{\Phi}_{M}=\mathbf{0}\]  
In this case (RC3 BP), the Jacobi integral is quite naturally present. It takes the form  
\[\frac{1}{2}{v_{x}}^{2}+\frac{1}{2}{v_{y}}^{2}-\frac{1}{2}{n_{S}}^{2}\Big{(}x^ {2}+y^{2}\Big{)}-\frac{\mu_{E}}{\sqrt{x^{2}+y^{2}}}-\frac{\mu_{S}}{{r_{S}}^{3} }\bigg{(}x^{2}-\frac{1}{2}y^{2}\bigg{)}-C_{J}=0 \tag{8}\]Further, according to the proposed approach of studying the perturbed SC motion in RC4 BP (2)-(4) on a basis of RC2 BP (5), it is proposed to use the method of \"variation of constants\" by [PERSON] [11]. As applied to the system of differential Equation (2a) and the set of the first integrals (6) for system (5), the corresponding equation of the method of variations can be represented as:  
\[\left(\frac{\partial\varphi_{j}}{\partial\mathbf{x}},\mathbf{\Phi}^{\Sigma} \right)+\left(\frac{\partial\varphi_{j}}{\partial\mathbf{C}},\frac{d}{dt} \mathbf{C}\right)=0,\ j=1\ldots 4 \tag{9}\]  
where  
\[ \begin{array}{l}\varphi_{j}(\mathbf{x},\mathbf{C})=0,\ j=1\ldots 4,\\ \varphi_{j}(\cdot),\frac{\partial\varphi_{j}}{\partial\mathbf{x}}(\cdot), \frac{\partial\varphi_{j}}{\partial\mathbf{C}}(\cdot)\in C\big{(}\mathbf{R}^{ 4}\times\mathbf{R}^{4}\backslash\left\{x,y=0\right\}\big{)},\\ \mathbf{x}=\big{(}v_{x},v_{y},x,y\big{)}^{\mathbf{T}},\ \mathbf{C}=\big{(}\Lambda_{ \xi},\Lambda_{\eta},\sigma,h\big{)}^{\mathbf{T}}\end{array} \]  
The perturbation vector (according to equations of the form (2a)):  
\[\mathbf{\Phi}^{\Sigma}=\left(\Phi_{M_{x}}+\Phi_{S_{x}},\Phi_{M_{y}}+\Phi_{S_{ y}},0,0\right)^{\mathbf{T}}\]  
Scalar expressions \(\varphi_{j}\left(\mathbf{x},\mathbf{C}\right)=0\) included in (9) are the first integrals (6) of system (5). By solving system (9) with respect to derivatives  
\[\frac{d}{dt}\Lambda_{\xi},\ \frac{d}{dt}\Lambda_{\eta},\ \frac{d}{dt}\sigma,\]  
Thus we can obtain differential equations describing the variation of the corresponding components of the vector \(\mathbf{C}\):  
\[\left. \begin{array}{l}\frac{d}{dt}\left(\begin{array}{c}\Lambda_{\xi}\\ \Lambda_{\eta}\end{array}\right)=A(t)\left(\begin{array}{c}\sigma\Big{(}\Phi _{M_{y}}+\Phi_{S_{y}}\Big{)}+\frac{d\sigma}{dt}\big{(}v_{y}+n_{S}x\Big{)}\\ -\sigma\big{(}\Phi_{M_{x}}+\Phi_{S_{x}}\big{)}-\frac{d\sigma}{dt}\big{(}v_{x}- n_{S}y\big{)}\end{array}\right)\\ \frac{d}{dt}\sigma=-y(\Phi_{M_{x}}+\Phi_{S_{x}})+x\Big{(}\[PERSON]_{M_{y}}+\[PERSON]_{S_{ y}}\Big{)}\end{array}