\(SL(2,{\mathbb {Z}})\) invariant action for probe (m, n) string in \(AdS_3\times S^3\times T^4\) with mixed three-form fluxes can be described by an integrable deformation of an one-dimensional Neumann–Rosochatius (NR) system. We present the deformed features of the integrable model and study general class of rotating and pulsating solutions by solving the integrable equations of motion. For the rotating string, the explicit solutions can be expressed in terms of elliptic functions. We make use of the integrals of motion to find out the scaling relation among conserved charges for the particular case of constant radii solutions. Then we study the closed (m, n) string pulsating in \(R_t\times S^3\). We find the string profile and calculate the total energy of such pulsating string in terms of oscillation number \(({\mathcal {N}})\) and angular momentum \(({\mathcal {J}})\).

具有混合三形式通量的\（AdS_3 \ times S ^ 3 \ times T ^ 4 \）中探针（m，  n）字符串的\（SL（2，{\ mathbb {Z}}）\）不变动作可以是用一维Neumann-Rosochatius（NR）系统的可积分形变描述。我们介绍了可积模型的变形特征，并通过求解运动的可积方程来研究一般类别的旋转和脉动解。对于旋转的弦，可以用椭圆函数表示显式解。我们利用运动积分来求出在恒定半径解的特殊情况下守恒电荷之间的比例关系。然后我们研究闭弦（m，  n）脉动\（R_t \ times S ^ 3 \）。我们找到弦的轮廓，并根据振动数\（（{{mathcal {N}}）\）和角动量\（（{\ mathcal {J}}）\）来计算这种脉动弦的总能量。